Algebra

Contents

Describe the term Logarithm 1

Describe the properties of Logarithm 2

Find the Logarithmic value of a number 3

Recall Sequences, Series, and Progressions 4

Recall Arithmetic Progressions and find the sum of n terms 5

Recall the Harmonic Progression 5

Discuss the Arithmetic and Harmonic Mean 6

Recall Geometric Progression 8

Find Sum of n-term and infinite term of a G.P. 9

Discuss Geometric Mean 10

Describe the term Partial Fraction 13

Find the Partial Fractions of: i. Non-Repeated Factors ii. Repeated Factors iii. Improper Fractions 13

Describe the Partial Fractions of Non-Reducible Quadratic Factors 13

Describe the Partial Fractions of Non-Reducible Quadratic Factors and Linear Factors 13

Describe fundamental principle of Counting 13

Describe the term Factorial 13

Describe Permutations 13

Define the term ⁿP_r 13

Describe Combinations 13

Define the term ⁿC_r 13

Find the number of ways using Permutations and Combinations Principles 13

State Binomial Theorem 13

Describe Pascal’s Triangle 13

Describe properties of the expansion of (x+a)ⁿ 13

Find independent term x, middle term, and the r^th term in the expansion of (x+a)ⁿ 13

Describe the term Logarithm

The term logarithm is a mathematical concept that represents the inverse operation of exponentiation. Given a base ‘b’ and a positive number ‘y’, the logarithm of ‘y’ to the base ‘b’ is defined as the power to which ‘b’ must be raised to obtain ‘y’. In other words, if we write ‘log_b(y) = x’, it means that ‘b^x = y’.

For example, consider the logarithm of 1000 to the base 10, denoted as log₁₀(1000). It can be written as log₁₀(1000) = 3, which means that 10 raised to the power 3 gives us 1000, i.e., 10³ = 1000.

The logarithm function is defined only for positive numbers, and its value is negative for values between 0 and 1, zero for the base value, and positive for values greater than 1. It is a useful concept in various fields, including mathematics, science, engineering, finance, and computer science.

The two most commonly used bases for logarithms are base 10 and base e, where e is the natural logarithm, an important mathematical constant approximately equal to 2.71828. The natural logarithm is denoted by the symbol ‘ln’, and the logarithm to the base 10 is denoted by the symbol ‘log’.

In addition to its applications in mathematical equations, logarithms are also used in measuring the intensity of sound and earthquake waves, and in comparing the brightness of stars.

Describe the properties of Logarithm

Logarithms have several properties that make them useful in various fields of mathematics and beyond. Some of the important properties of logarithms are:

1. Product rule: The logarithm of a product of two numbers is equal to the sum of their logarithms. That is, log_b(xy) = log_b(x) + log_b(y). For example, log₂ (8*16) = log₂(8) + log₂(16) = 3 + 4 = 7.
2. Quotient rule: The logarithm of a quotient of two numbers is equal to the difference of their logarithms. That is, log_b(x/y) = log_b(x) – log_b(y). For example, log₁₀(100/4) = log₁₀(100) – log₁₀(4) = 2 – 0.602 = 1.398.
3. Power rule: The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. That is, log_b(x^y) = y * log_b(x). For example, log₂(16³) = 3log₂(16) = 34 = 12.
4. Change of base rule: The logarithm of a number to a certain base can be expressed in terms of the logarithm of the same number to another base. That is, log_a(x) = log_b(x) / log,(a), where ‘a’ and ‘b’ are two different bases. For example, log₃(81) = log₁₀(81) / log₁₀(3) = 1.908 / 0.477 = 4.
5. Inverse property: The logarithm of a number and its exponential function are inverse functions of each other. That is, if y = log_b(x), then x = b^y. For example, if log₂(16) = 4, then 2⁴ = 16.
6. Logarithm of 1: The logarithm of 1 to any base is always zero. That is, log_b(1) = 0.
7. Logarithm of base: The logarithm of the base itself is always equal to 1. That is, log_b(b) = 1.

These properties make logarithms a powerful tool in solving equations and simplifying mathematical expressions. They are also useful in data analysis, signal processing, and scientific modelling, among other applications.

Find the Logarithmic value of a number

To find the logarithmic value of a number, we use a logarithmic function, which is the inverse of an exponential function. Logarithms are used to represent the power to which a base must be raised to produce a given number. The most common bases used in logarithmic functions are 10, e, and 2.

For example, if we want to find the logarithmic value of a number x to the base 10, we write it as log₁₀(x) or simply log(x). If we want to find the logarithmic value of the same number to the base e, we write it as ln(x) or natural logarithm of x. Similarly, if we want to find the logarithmic value of the same number to the base 2, we write it as log₂(x).

To find the logarithmic value of a number using a calculator or software, we simply enter the number and its base as an argument of the logarithmic function. For example, to find the logarithmic value of 1000 to the base 10, we can write log(1000) in a calculator, which gives the result as 3. Similarly, to find the logarithmic value of 7 to the base 2, we can write log₂(7) in a calculator, which gives the result as 2.807.

We can also use logarithmic properties to simplify the computation of logarithmic values of complex expressions. For example, if we want to find the logarithmic value of the expression (4*5)/(2³) to the base 10, we can use the properties of logarithms as follows:

log₁₀[(45)/(2³)] = log₁₀(45) – log₁₀(2³) (using quotient and product rules)

= log₁₀(4) + log₁₀(5) – 3*log₁₀(2) (using product and power rules)

= 0.602 + 0.699 – 0.903

= 0.398

Therefore, the logarithmic value of the expression (4*5)/(2³) to the base 10 is 0.398.

Recall Sequences, Series, and Progressions

Sequences, series, and progressions are important concepts in mathematics, particularly in calculus, analysis, and number theory.

A sequence is a list of numbers arranged in a specific order. The individual numbers in a sequence are called terms. The sequence can be finite or infinite. For example, 1, 3, 5, 7, 9 is a finite sequence of five terms, while 1, 2, 3, 4, … is an infinite sequence that goes on indefinitely.

A series is the sum of the terms in a sequence. If the sequence is finite, the series is also finite. However, if the sequence is infinite, the series may be either finite or infinite. For example, the series 1 + 3 + 5 + 7 + 9 is the sum of the terms in the finite sequence 1, 3, 5, 7, 9, and has a finite value of 25. On the other hand, the series 1 + 2 + 3 + 4 + … is the sum of the terms in the infinite sequence 1, 2, 3, 4, … and has an infinite value.

A progression is a sequence in which the difference between successive terms is constant. There are different types of progressions, including arithmetic, geometric, and harmonic progressions.

An arithmetic progression (AP) is a sequence in which the difference between successive terms is constant. The constant difference is called the common difference. For example, 1, 3, 5, 7, 9 is an arithmetic progression with a common difference of 2.

A geometric progression (GP) is a sequence in which each term is a constant multiple of the previous term. The constant ratio is called the common ratio. For example, 1, 2, 4, 8, 16 is a geometric progression with a common ratio of 2.

A harmonic progression (HP) is a sequence in which each term is the reciprocal of a corresponding term in an arithmetic progression. For example, 1/2, 1/3, 1/4, 1/5, 1/6 is a harmonic progression obtained from the arithmetic progression 2, 3, 4, 5, 6.

These concepts are widely used in mathematics and its applications, such as physics, engineering, and computer science.

Recall Arithmetic Progressions and find the sum of n terms

Arithmetic Progression (AP) is a sequence of numbers in which each term is obtained by adding a fixed number to the previous term. This fixed number is called the common difference (d). For example, 2, 5, 8, 11, 14 is an AP with a common difference of 3.

The sum of the first n terms of an AP is given by the formula:

S_n = (n/2) * [2a + (n-1)d]

Where Sn is the sum of the first n terms of the AP, a is the first term of the AP and d is the common difference.

For example, if we have the AP 3, 6, 9, 12, 15, 18, and we want to find the sum of the first 4 terms, we can use the formula:

S₄ = (4/2) * [2(3) + (4-1)(3)]

S₄ = 2 * (6 + 9)

S₄ = 30

Therefore, the sum of the first 4 terms of the AP is 30.

Recall the Harmonic Progression

A Harmonic Progression (HP) is a sequence of numbers in which each term is obtained by taking the reciprocal of the corresponding term in an Arithmetic Progression (AP). For example, if we have the AP 2, 4, 6, 8, then the corresponding HP is 1/2, 1/4, 1/6, 1/8.

The nth term of the HP is given by:

a_n = 1/(a + (n-1)d)

Where a is the first term of the AP, d is the common difference of the AP, and n is the term number.

The sum of the first n terms of the HP is given by:

S_n = (n/(a + (n-1)d))

For example, if we have the AP 1, 2, 3, 4 and we want to find the sum of the first 4 terms of the corresponding HP, we can first find the corresponding HP by taking the reciprocal of the terms of the AP:

1, 1/2, 1/3, 1/4

Then we can use the formula for the sum of the first n terms of the HP:

S₄ = (4/(1 + (4-1)(1)))

S₄ = 4/4

S₄ = 1

Therefore, the sum of the first 4 terms of the HP is 1.

Discuss the Arithmetic and Harmonic Mean

Arithmetic Mean:

The arithmetic mean (AM) is a measure of central tendency and is calculated as the sum of all the values in a dataset divided by the number of values. In other words, it is the average value of the dataset.

If we have a set of n values (a₁, a₂, …, a_n), then the arithmetic mean is given by:

AM = (a₁ + a₂ + … + a_n) / n

For example, if we have the set of values 2, 4, 6, 8, the arithmetic mean is:

AM = (2 + 4 + 6 + 8) / 4 = 5

Harmonic Mean:

The harmonic mean (HM) is a measure of central tendency and is calculated as the reciprocal of the arithmetic mean of the reciprocals of the values in a dataset. In other words, it is the average of the reciprocals of the values in the dataset.

If we have a set of n values (a₁, a₂, …, a_n), then the harmonic mean is given by:

HM = n / ((1/a₁) + (1/a₂) + … + (1/a_n))

For example, if we have the set of values 2, 4, 6, 8, the harmonic mean is:

HM = 4 / ((1/2) + (1/4) + (1/6) + (1/8)) ≈ 3.43

Comparison:

The arithmetic mean is generally greater than or equal to the harmonic mean, with equality only when all the values are the same. For a set of positive values, the harmonic mean is the appropriate measure of central tendency when the values are rates or ratios, while the arithmetic mean is the appropriate measure for non-ratio variables.

For example, if we have the set of values 2, 4, 6, 8, the arithmetic mean is 5 and the harmonic mean is 3.43. In this case, the arithmetic mean is greater than the harmonic mean, which is typical for most datasets. However, if the values represent rates or ratios, such as miles per gallon or price-to-earnings ratios, the harmonic mean would be a more appropriate measure of central tendency.

Recall Geometric Progression

Geometric progression (GP) is a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by a constant factor called the common ratio (r). The general form of a GP is: a, ar, ar², ar³, … , ar^(n-1) where a is the first term, r is the common ratio, and n is the number of terms in the sequence.

Properties of Geometric Progression:

1. The nth term of a GP can be represented as: ar^(n-1)
2. The sum of the first n terms of a GP can be represented as: S_n = a(1 – rⁿ)/(1 – r) (when r is not equal to 1)
3. The sum of an infinite GP can be represented as: S = a/(1 – r) (when |r| < 1)
4. The common ratio (r) is calculated by dividing any term in the GP by the preceding term.
5. If the common ratio is greater than 1, the GP is increasing and if it is less than 1, the GP is decreasing.

Example:

Find the sum of the first five terms of the GP whose first term is 2 and the common ratio is 3.

Solution:

a = 2, r = 3, n = 5

Using the formula for the sum of the first n terms of a GP:

S₅ = a(1 – rⁿ)/(1 – r)

S₅ = 2(1 – 3⁵)/(1 – 3)

S₅ = 242

Therefore, the sum of the first five terms of the given GP is 242.

Find Sum of n-term and infinite term of a G.P.

A geometric progression is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed number called the common ratio. A geometric progression can be expressed as a, ar, ar², ar³, … , ar^n-1, …, where a is the first term, r is the common ratio, and n is the number of terms.

Finding the sum of n-terms of a geometric progression

The sum of the first n terms of a geometric progression is given by the formula:

S_n = a(1 – rⁿ) / (1 – r)

where a is the first term, r is the common ratio, and n is the number of terms.

For example, consider the geometric progression 1, 2, 4, 8, 16, …. The first term a = 1, and the common ratio r = 2. Suppose we want to find the sum of the first 5 terms of this sequence. Using the formula, we get:

S₅ = 1(1 – 2⁵) / (1 – 2) = 31

Therefore, the sum of the first 5 terms of the sequence 1, 2, 4, 8, 16, … is 31.

Finding the infinite sum of a geometric progression

If the common ratio of a geometric progression is between -1 and 1, then the sum of the infinite terms of the sequence can be found using the formula:

S = a / (1 – r)

For example, consider the geometric progression 1/2, 1/4, 1/8, 1/16, …. The first term a = 1/2, and the common ratio r = 1/2. Since the common ratio is between -1 and 1, we can use the formula to find the sum of the infinite terms of the sequence:

S = (1/2) / (1 – 1/2) = 1

Therefore, the sum of the infinite terms of the sequence 1/2, 1/4, 1/8, 1/16, … is 1.

Discuss Geometric Mean

Geometric Mean is a measure of central tendency which is calculated by multiplying a set of values and taking the nth root of the product. It is often used to measure the average rate of change or growth. In this learning outcome, we will discuss geometric mean in detail along with suitable examples.

Definition of Geometric Mean:

Geometric mean is defined as the nth root of the product of n numbers. Mathematically, it can be expressed as:

GM = (X₁ * X₂ * X₃ * … * X_n) ^(1/n)

where GM is the geometric mean, X₁, X₂, X_3, …, x_n are the n numbers whose geometric mean is to be calculated.

Example 1: Calculating Geometric Mean of Two Numbers

Suppose we have two numbers, 4 and 16. The geometric mean of these two numbers can be calculated as:

GM = (4 * 16) ^(1/2) = 8

Thus, the geometric mean of 4 and 16 is 8.

Example 2: Calculating Geometric Mean of Three Numbers

Suppose we have three numbers, 2, 4 and 8. The geometric mean of these three numbers can be calculated as:

GM = (2 * 4 * 8) ^(1/3) = 4

Thus, the geometric mean of 2, 4 and 8 is 4.

Example 3: Applications of Geometric Mean in Finance

Geometric mean is often used in finance to measure the average rate of return on an investment. Suppose an investment has the following returns over a period of 3 years:

Year 1: 10%

Year 2: 20%

Year 3: 30%

The average rate of return can be calculated using the arithmetic mean as:

AM = (10 + 20 + 30)/3 = 20%

However, this method assumes that the returns are independent of each other, which may not be the case in real life. Therefore, the geometric mean is used to calculate the average rate of return as:

GM = (1.1 * 1.2 * 1.3)^(1/3) – 1 = 21.54%

Thus, the geometric mean of the returns is 21.54%.

Example 4: Geometric Mean in Biology

Geometric mean is also used in biology to calculate the mean growth rate of a population. Suppose a population grows at the following rates over a period of 4 years:

Year 1: 10%

Year 2: 20%

Year 3: 30%

Year 4: 40%

The mean growth rate can be calculated using the geometric mean as:

GM = (1.1 * 1.2 * 1.3 * 1.4) ^(1/4) – 1 = 23.46%

Thus, the mean growth rate of the population is 23.46%.

Conclusion:

Geometric mean is a useful measure of central tendency which is used to calculate the average rate of change or growth. It is particularly useful when dealing with percentages or ratios. Geometric means has various applications in fields such as finance, biology, and statistics.

Describe the term Partial Fraction

Partial fractions is a technique used to decompose a rational function into a sum of simpler fractions. In this learning outcome, we will discuss partial fractions in detail along with suitable examples.

Definition of Partial Fractions:

A partial fraction is a fraction in which the numerator is a polynomial of lesser degree than the denominator. A rational function can be expressed as a sum of partial fractions by decomposing the denominator into a product of linear and irreducible quadratic factors.

Example 1: Simple Partial Fraction Decomposition

Suppose we have the rational function:

F(x) = (x + 2)/(x² + 5x + 6)

We can decompose the denominator into a product of linear factors as:

x² + 5x + 6 = (x + 3)(x + 2)

The partial fraction decomposition of F(x) can be written as:

F(x) = A/(x + 3) + B/(x + 2)

where A and B are constants to be determined. To solve for A and B, we can use the method of equating coefficients. Multiplying both sides by the denominator, we get:

(x + 2)A + (x + 3)B = x + 2

Substituting x = -3, we get:

A = -1

Substituting x = -2, we get:

B = 3

Thus, the partial fraction decomposition of F(x) is:

F(x) = -1/(x + 3) + 3/(x + 2)

Example 2: Partial Fraction Decomposition of an Irreducible Quadratic Factor

Suppose we have the rational function:

F(x) = (x² + 2x + 3)/(x² + x + 1)(x – 2)

We can decompose the denominator into a product of linear and irreducible quadratic factors as:

x² + x + 1 = (x + (1/2) + i(sqrt(3)/2))(x + (1/2) – i(sqrt(3)/2))

The partial fraction decomposition of F(x) can be written as:

F(x) = (Ax + B)/(x² + x + 1) + C/(x – 2)

where A, B and C are constants to be determined. To solve for A, B and C, we can use the method of equating coefficients. Multiplying both sides by the denominators, we get:

Ax² + (A + B + C)x + (B – 2A) = x² + 2x + 3

Equating coefficients of like terms, we get:

A = 1/3

B = 5/3

C = -4/3

Thus, the partial fraction decomposition of F(x) is:

F(x) = (x/3 + 5/3)/(x² + x + 1) – 4/(x – 2)

Example 3: Partial Fraction Decomposition with Repeated Factors

Suppose we have the rational function:

F(x) = (3x² – 2x + 1)/(x – 1)³

The partial fraction decomposition of F(x) can be written as:

F(x) = A/(x – 1) + B/(x – 1)² + C/(x – 1)³

where A, B and C are constants to be determined. To solve for A, B and C, we can use the method of equating coefficients.

Find the Partial Fractions of: i. Non-Repeated Factors ii. Repeated Factors iii. Improper Fractions

In this learning outcome, we will discuss three different cases for finding partial fractions – i) non-repeated factors, ii) repeated factors, and iii) improper fractions. We will provide detailed explanations along with examples for each case.

i) Partial Fractions of Non-Repeated Factors:

A non-repeated factor is a factor that appears only once in the denominator of the given rational function. The partial fraction decomposition of a rational function with non-repeated factors can be found by equating the coefficients of the partial fraction decomposition with the coefficients of the original function.

Example 1: Partial Fraction Decomposition of a Function with Non-Repeated Factors

Consider the rational function:

F(x) = (x + 2)/(x² + 3x + 2)

We can factor the denominator as (x+1)(x+2). Therefore, the partial fraction decomposition of F(x) is given by:

F(x) = A/(x+1) + B/(x+2)

To determine the values of A and B, we equate the coefficients of the partial fraction decomposition to the coefficients of the original function. This gives us:

A(x+2) + B(x+1) = x+2

Solving for A and B, we get:

A = 1 and B = -1

Therefore, the partial fraction decomposition of F(x) is:

F(x) = 1/(x+1) – 1/(x+2)

ii) Partial Fractions of Repeated Factors:

A repeated factor is a factor that appears more than once in the denominator of the given rational function. The partial fraction decomposition of a rational function with repeated factors can be found by expressing it in terms of simpler fractions and then equating the coefficients.

Example 2: Partial Fraction Decomposition of a Function with Repeated Factors

Consider the rational function:

F(x) = (x + 1)/(x-1)³

To find the partial fraction decomposition of F(x), we express it as:

F(x) = A/(x-1) + B/(x-1)² + C/(x-1)³

To determine the values of A, B, and C, we equate the coefficients of the partial fraction decomposition to the coefficients of the original function. This gives us:

A(x-1)² + B(x-1) + C = x+1

Solving for A, B, and C, we get:

A = 1/2, B = 0, and C = 3/2

Therefore, the partial fraction decomposition of F(x) is:

F(x) = 1/2(x-1) + 3/2(x-1)³

iii) Partial Fractions of Improper Fractions:

An improper fraction is a fraction in which the degree of the numerator is greater than or equal to the degree of the denominator. To find the partial fraction decomposition of an improper fraction, we first divide the numerator by the denominator to obtain a polynomial plus a fraction with a proper fraction.

Example 3: Partial Fraction Decomposition of an Improper Fraction

Consider the rational function:

F(x) = (3x² + 2x + 1)/(x² + x + 1)

Since the degree of the numerator is greater than the degree of the denominator, we perform the division as:

F(x) = 3x – 1 + (4x + 1)/(x² + x + 1)

The partial fraction decomposition of the second term can be found by completing the square and expressing it in terms of simpler fractions.

Describe the Partial Fractions of Non-Reducible Quadratic Factors

In this learning outcome, we will discuss how to find the partial fraction decomposition of a rational function with non-reducible quadratic factors in the denominator. A non-reducible quadratic factor is a quadratic expression that cannot be factored into linear factors with real coefficients.

To find the partial fraction decomposition of a rational function with non-reducible quadratic factors, we express the quadratic factor in terms of a linear term and a constant term using the technique of completing the square. Then, we express the original rational function as a sum of fractions with denominators that are linear terms and a constant term. The coefficients of the partial fractions can be found by equating the coefficients of the original function with the coefficients of the partial fraction decomposition.

Example: Partial Fraction Decomposition of a Function with Non-Reducible Quadratic Factors

Consider the rational function:

F(x) = (2x + 1)/(x² + x + 2)

The denominator of this rational function is a non-reducible quadratic factor. To find the partial fraction decomposition of F(x), we first express the quadratic factor in terms of a linear term and a constant term. Completing the square, we get:

x² + x + 2 = (x + 1/2)² + 7/4

Substituting this expression into the denominator of F(x), we get:

F(x) = (2x + 1)/[(x + 1/2)² + 7/4]

We can now express F(x) in terms of partial fractions:

F(x) = A/(x + 1/2) + B/[(x + 1/2)² + 7/4]

To determine the values of A and B, we equate the coefficients of the partial fraction decomposition with the coefficients of the original function. This gives us:

A[(x + 1/2)² + 7/4] + B(x + 1/2) = 2x + 1

Solving for A and B, we get:

A = (4x + 3)/(3(x + 1/2)² + 7) and B = (-2x – 1)/(3(x + 1/2)² + 7)

Therefore, the partial fraction decomposition of F(x) is:

F(x) = (4x + 3)/(3(x + 1/2)² + 7) – (2x + 1)/(3(x + 1/2)² + 7) * (x + 1/2)

Describe the Partial Fractions of Non-Reducible Quadratic Factors and Linear Factors

Partial fraction decomposition is a process of breaking down a complex rational function into simpler fractions that can be easily integrated or manipulated. In this learning outcome, we will focus on partial fractions of non-reducible quadratic factors and linear factors.

1. Partial Fractions of Non-Reducible Quadratic Factors

A non-reducible quadratic factor is a quadratic expression that cannot be factored into linear factors with real coefficients. The general form of such a quadratic factor is (ax² + bx + c), where a, b, and c are real numbers and a is not equal to zero. To write a rational function with a non-reducible quadratic factor in partial fraction form, we use the following steps:

Step 1: Factor the quadratic expression (ax² + bx + c) into irreducible quadratic factors, for example:

(ax² + bx + c) = (mx + n)(px² + qx + r), where m, n, p, q, and r are constants.

Step 2: Write the partial fraction form for each irreducible quadratic factor, for example:

(px² + qx + r) = (Ax + B)/(ax² + bx + c), where A and B are constants.

Step 3: Combine the partial fraction forms into a single expression, for example:

(ax² + bx + c)/(mx + n)(px² + qx + r) = (A₁x + B₁)/(mx + n) + (A₂x + B₂)/(ax² + bx + c)

Example:

Consider the following rational function: (3x² + 4x + 1)/(x² + x – 6).

We factor the denominator as (x + 3)(x – 2), then write the partial fractions as:

(3x² + 4x + 1)/(x² + x – 6) = A/(x + 3) + B/(x – 2), where A and B are constants.

To find the values of A and B, we multiply both sides of the equation by (x + 3)(x – 2) and simplify to obtain:

3x² + 4x + 1 = A(x – 2) + B(x + 3)

Substituting x = -3 and x = 2 into this equation gives a system of two equations in two variables, which can be solved to find A = 1 and B = 2. Therefore, the partial fraction form of the original rational function is:

(3x² + 4x + 1)/(x² + x – 6) = 1/(x + 3) + 2/(x – 2)

1. Partial Fractions of Linear Factors

A linear factor is a polynomial expression of the form (ax + b), where a and b are constants. To write a rational function with a linear factor in partial fraction form, we use the following steps:

Step 1: Write the linear factor in the form of (ax + b).

Step 2: Write the partial fraction form for the linear factor, for example:

(ax + b) = A/(ax + b), where A is a constant.

Describe fundamental principle of Counting

The fundamental principle of counting is a basic concept in combinatorics that provides a systematic way of counting the number of possible outcomes of an event. It is an important tool for solving a variety of problems related to probability, statistics, and other areas of mathematics. In this learning outcome, we will discuss the fundamental principle of counting and its various applications.

The Fundamental Principle of Counting

The fundamental principle of counting states that if an event can occur in m ways and a second event can occur in n ways, then the two events can occur in m × n ways. In other words, the total number of possible outcomes for the two events is the product of the number of outcomes for each event. This principle can be extended to any number of events.

Example:

Suppose you have three shirts, two pants, and two pairs of shoes. How many different outfits can you make by selecting one shirt, one pair of pants, and one pair of shoes?

Solution:

According to the fundamental principle of counting, the total number of possible outfits is equal to the product of the number of choices for each item. Therefore, the number of possible outfits is 3 × 2 × 2 = 12.

Applications of the Fundamental Principle of Counting

The fundamental principle of counting can be used to solve a variety of problems involving counting and probability. Some of the common applications of the principle are:

a. Permutations and Combinations:

The principle can be used to determine the number of permutations and combinations of a set of objects.

For example, the number of ways to arrange n distinct objects in a row is n! (n factorial), which is equal to the product of all positive integers from 1 to n. Similarly, the number of ways to select k objects from a set of n distinct objects without regard to order is denoted by nCk (n choose k), which is equal to n!/(k!(n-k)!).

b. Probability:

The principle can be used to determine the probability of an event. For example, if the probability of event A occurring is p and the probability of event B occurring is q, then the probability of both events occurring is p × q. Similarly, the probability of either event A or event B occurring is given by p + q – p × q.

c. Counting Problems:

The principle can be used to solve various counting problems, such as the number of ways to arrange objects with certain restrictions, the number of paths in a grid, or the number of ways to color a map. These problems can be solved by breaking them down into smaller, more manageable parts and applying the fundamental principle of counting.

In conclusion, the fundamental principle of counting is an important concept in combinatorics that provides a systematic way of counting the number of possible outcomes of an event. It has a wide range of applications in various areas of mathematics and can be used to solve a variety of counting and probability problems.

Describe the term Factorial

In mathematics, the factorial is a mathematical operation that is used to calculate the product of all positive integers from 1 to a given positive integer. The factorial is denoted by the exclamation mark (!) symbol. In this learning outcome, we will discuss the factorial and its various properties and applications.

Definition of Factorial

The factorial of a positive integer n, denoted by n!, is the product of all positive integers from 1 to n. That is, n! = 1 × 2 × 3 × … × (n-1) × n. For example, 5! = 1 × 2 × 3 × 4 × 5 = 120.

Properties of Factorial

The factorial has several properties that are useful in various areas of mathematics. Some of the main properties are:

a. 0! = 1: The factorial of 0 is defined to be 1. This is a convention that is used in combinatorics and other areas of mathematics.

b. n! = n × (n-1)!: The factorial of a positive integer n is equal to n times the factorial of (n-1). This property can be used to simplify calculations involving factorials.

c. (n+1)! = (n+1) × n!: The factorial of (n+1) is equal to (n+1) times the factorial of n. This property is a special case of property b.

d. n!/(n-k)! = n × (n-1) × … × (n-k+1): This formula is used to calculate the number of ways to select k objects from a set of n distinct objects without regard to order. The notation for this is denoted by nCk (n choose k).

Applications of Factorial

The factorial has several applications in mathematics and other areas. Some of the main applications are:

a. Combinatorics:

The factorial is used to calculate the number of permutations and combinations of a set of objects. For example, the number of ways to arrange n distinct objects in a row is n! (n factorial). Similarly, the number of ways to select k objects from a set of n distinct objects without regard to order is denoted by nCk (n choose k), which is equal to n!/(k!(n-k)!).

b. Probability:

The factorial is used to calculate the probability of an event in which order matters. For example, the probability of selecting two cards from a deck of cards in which order matters is equal to (52!/50!) / ⁵²C₂, which simplifies to 1/26.

c. Number Theory:

The factorial is used in number theory to calculate various properties of numbers, such as the number of divisors of a number, the number of ways to factorize a number, and the value of certain mathematical functions, such as the gamma function.

In conclusion, the factorial is a mathematical operation that is used to calculate the product of all positive integers from 1 to a given positive integer. It has several properties and applications in various areas of mathematics, such as combinatorics, probability, and number theory. The factorial is an important tool in mathematics and is used in many different areas of study.

Describe Permutations

In mathematics, permutations are a way of arranging objects in a specific order. In this learning outcome, we will discuss permutations, how they are calculated, and their applications.

Definition of Permutations

A permutation is an arrangement of objects in a specific order. For example, if we have three objects A, B, and C, the six possible permutations are ABC, ACB, BAC, BCA, CAB, and CBA. Permutations are often denoted by the symbol P(n, r), where n is the total number of objects and r is the number of objects being arranged.

Calculation of Permutations

The number of permutations of n objects taken r at a time, denoted by P(n, r), is given by the formula:

P(n, r) = n! / (n – r)!

where n! is the factorial of n. For example, the number of permutations of 5 objects taken 3 at a time is:

P(5, 3) = 5! / (5 – 3)! = 5! / 2! = (5 × 4 × 3) / (2 × 1) = 60

Properties of Permutations

Permutations have several properties that are useful in various areas of mathematics. Some of the main properties are:

a. P(n, n) = n!: The number of permutations of n objects taken n at a time is equal to n factorial.

b. P(n, 0) = 1: The number of permutations of n objects taken 0 at a time is equal to 1.

c. P(n, r) = P(n-1, r) + P(n-1, r-1):

The number of permutations of n objects taken r at a time is equal to the sum of the number of permutations of n-1 objects taken r at a time and the number of permutations of n-1 objects taken r-1 at a time.

Applications of Permutations

Permutations have several applications in mathematics and other areas. Some of the main applications are:

a. Combinatorics:

Permutations are used to calculate the number of ways to arrange objects in a specific order. For example, the number of ways to arrange the letters in the word “MISSISSIPPI” is 11! / (4! × 4! × 2!).

b. Probability:

Permutations are used to calculate the probability of an event in which order matters. For example, the probability of selecting two cards from a deck of cards in which order matters is equal to P(52, 2) / 52! = 1/1326.

c. Group Theory:

Permutations are used in group theory to describe the symmetry of objects. For example, the permutation group of a cube consists of all possible permutations of the vertices of the cube.

In conclusion, permutations are a way of arranging objects in a specific order. They have several properties and applications in various areas of mathematics, such as combinatorics, probability, and group theory. The calculation of permutations is an important tool in mathematics and is used in many different areas of study.

Define the term ⁿP_r

In this learning outcome, we will define the term ⁿP_r, which represents the number of permutations of n objects taken r at a time.

Definition of ⁿP_r

ⁿP_r is a mathematical notation used to represent the number of permutations of n objects taken r at a time. The notation is read as “n permute r” or ⁿP_r. It is calculated using the formula:

ⁿP_r = n! / (n-r)!

where n is the total number of objects, and r is the number of objects being arranged.

Example

Suppose we have a set of 6 letters {A, B, C, D, E, F}. We want to know the number of ways we can arrange 3 letters from this set. In this case, n = 6 and r = 3. Using the formula for permutations, we get:

⁶P₃= 6! / (6-3)! = 6! / 3! = (6 × 5 × 4) / (3 × 2 × 1) = 120

Therefore, there are 120 ways to arrange 3 letters from the set {A, B, C, D, E, F}.

Properties of ⁿP_r

ⁿP_r has several properties that are useful in various areas of mathematics. Some of the main properties are:

a. ⁿP_n = n!: The number of permutations of n objects taken n at a time is equal to n factorial.

b. ⁿP₀ = 1: The number of permutations of n objects taken 0 at a time is equal to 1.

c. ⁿP_r = ^n-1P_r + ^n-1P_r-1: The number of permutations of n objects taken r at a time is equal to the sum of the number of permutations of n-1 objects taken r at a time and the number of permutations of n-1 objects taken r-1 at a time.

Applications of ⁿP_r

ⁿP_r has several applications in mathematics and other areas. Some of the main applications are:

a. Combinatorics:

ⁿP_r is used to calculate the number of ways to arrange objects in a specific order. For example, the number of ways to select a committee of 3 people from a group of 6 is ⁶P₃.

b. Probability:

ⁿP_r is used to calculate the probability of an event in which order matters. For example, the probability of selecting 3 cards from a deck of cards in which order matters is equal to ⁵²P₃ / 52! = 1/22100.

Describe Combinations

In this learning outcome, we will define combinations, which is a way of selecting a group of objects without regard to order.

Definition of Combinations

A combination is a way of selecting a group of objects from a larger set without regard to order. In other words, a combination is a selection of r objects from a set of n objects without considering the order in which they are selected. The number of possible combinations of r objects from a set of n objects is denoted by C(n, r) or nCr and is given by the formula:

C(n, r) = n! / (r! * (n-r)!)

where n is the total number of objects and r is the number of objects being selected.

Example

Suppose we have a set of 5 letters {A, B, C, D, E}. We want to know the number of ways we can select 3 letters from this set. In this case, n = 5 and r = 3. Using the formula for combinations, we get:

C(5, 3) = 5! / (3! * (5-3)!) = (5 × 4 × 3) / (3 × 2 × 1) = 10

Therefore, there are 10 possible combinations of 3 letters from the set {A, B, C, D, E}.

Properties of Combinations

Combinations have several properties that are useful in various areas of mathematics. Some of the main properties are:

a. C(n, r) = C(n, n-r): The number of combinations of r objects from a set of n objects is equal to the number of combinations of n-r objects from the same set.

b. C(n, r) = C(n-1, r) + C(n-1, r-1): The number of combinations of r objects from a set of n objects is equal to the sum of the number of combinations of r objects from a set of n-1 objects and the number of combinations of r-1 objects from a set of n-1 objects.

c. C(n, 0) = 1 and C(n, n) = 1: The number of combinations of 0 objects from a set of n objects is equal to 1, and the number of combinations of n objects from the same set is also equal to 1.

Applications of Combinations

Combinations have several applications in mathematics and other areas. Some of the main applications are:

a. Probability:

Combinations are used to calculate the probability of an event in which order does not matter. For example, the probability of selecting 3 cards from a deck of cards in which order does not matter is equal to C(52, 3) / C(52, 3) = 1/22100.

b. Combinatorics:

Combinations are used to calculate the number of ways to select a group of objects from a larger set without regard to order. For example, the number of ways to select a committee of 3 people from a group of 6 is C(6, 3).

c. Computer Science:

Combinations are used in computer science algorithms to generate all possible combinations of a set of objects. This is used in various applications such as password cracking, data analysis, and machine learning.

Define the term ⁿC_r

In this learning outcome, we will define binomial coefficient, which is a way of counting the number of combinations of r objects that can be selected from a set of n distinct objects.

Definition of Binomial Coefficient

The binomial coefficient, denoted as C(n, r) or nCr, represents the number of ways to choose r objects from a set of n distinct objects, without replacement and without regard to the order in which the objects are chosen. The formula for the binomial coefficient is given as:

C(n, r) = n! / (r! * (n-r)!)

where n is the total number of objects, and r is the number of objects to be selected.

Example

Suppose we have a set of 5 letters {A, B, C, D, E}. We want to know the number of ways we can select 3 letters from this set. In this case, n = 5 and r = 3. Using the formula for binomial coefficient, we get:

C(5, 3) = 5! / (3! * (5-3)!) = (5 × 4 × 3) / (3 × 2 × 1) = 10

Therefore, there are 10 possible ways to select 3 letters from the set {A, B, C, D, E}.

Properties of Binomial Coefficient

Binomial coefficients have several properties that are useful in various areas of mathematics. Some of the main properties are:

a. Symmetry:

C(n, r) = C(n, n-r) for all values of n and r.

b. Pascal’s Identity:

C(n, r) = C(n-1, r) + C(n-1, r-1) for all values of n and r.

c. Recursive Formula:

C(n, r) can be calculated recursively as C(n, r) = C(n-1, r) + C(n-1, r-1) with initial conditions of C(n, 0) = 1 and C(n, n) = 1.

Applications of Binomial Coefficient

Binomial coefficients have several applications in mathematics and other areas. Some of the main applications are:

a. Probability:

Binomial coefficient is used to calculate the probability of an event that can have only two possible outcomes, such as success or failure. For example, the probability of getting exactly 2 heads in 5 flips of a fair coin is given by C(5, 2) × (1/2)⁵ = 10/32.

b. Combinatorics:

Binomial coefficient is used to count the number of ways to select a group of objects from a larger set without regard to order. For example, the number of ways to select a committee of 3 people from a group of 6 is C(6, 3).

c. Algebra:

Binomial coefficient is used in algebra to expand the binomial expression (a + b)ⁿ. The coefficients of the expanded terms are given by the binomial coefficients C(n, r). For example, (a + b)² = a² + 2ab + b², where the coefficients 1, 2, and 1 are given by the binomial coefficients C(2, 0), C(2, 1), and C(2, 2), respectively.

Find the number of ways using Permutations and Combinations Principles

In this learning outcome, we will learn how to apply the principles of permutations and combinations to solve problems related to counting the number of ways of selecting or arranging objects.

Permutations

Permutations are arrangements of objects in a specific order. The number of permutations of r objects selected from a set of n distinct objects is given by the formula:

ⁿP_r = n! / (n-r)!

where n is the total number of objects, and r is the number of objects to be selected.

Example: How many ways can 3 different books be arranged on a shelf?

Solution: Here, n = 3 (number of books), and r = 3 (number of books to be arranged). Using the formula for permutations, we get:

³P₃ = 3! / (3-3)! = 6

Therefore, there are 6 ways in which 3 different books can be arranged on a shelf.

Combinations

Combinations are selections of objects without any specific order. The number of combinations of r objects selected from a set of n distinct objects is given by the formula:

ⁿC_r = n! / (r! * (n-r)!)

where n is the total number of objects, and r is the number of objects to be selected.

Example: How many ways can 3 different books be selected from a set of 5 different books?

Solution: Here, n = 5 (total number of books), and r = 3 (number of books to be selected). Using the formula for combinations, we get:

⁵C₃ = 5! / (3! * (5-3)!) = 10

Therefore, there are 10 ways in which 3 different books can be selected from a set of 5 different books.

Using Permutations and Combinations Principles

Sometimes, problems require the use of both permutations and combinations principles. In such cases, we need to identify whether we are looking for an arrangement (permutation) or a selection (combination), and then apply the appropriate formula.

Example: How many ways can 4 different books be arranged on a shelf if 2 of them must be together?

Solution: We can first select the 2 books that must be together in ²C₂ = 1 way. We can then arrange these 2 books in 2! = 2 ways. We can arrange the remaining 2 books in 2! = 2 ways. Therefore, the total number of arrangements is:

1 * 2! * 2! = 4

Therefore, there are 4 ways to arrange 4 different books on a shelf if 2 of them must be together.

In conclusion, by understanding and applying the principles of permutations and combinations, we can solve various counting problems that involve selecting or arranging objects. It is important to identify the type of problem and apply the appropriate formula to obtain the correct solution.

State Binomial Theorem

The binomial theorem is an important formula that allows us to expand the power of a binomial expression, that is, an expression consisting of two terms, raised to a positive integer power. The theorem provides a general formula for finding the coefficients of each term in the expansion.

The binomial theorem states that for any positive integer n:

(a + b)ⁿ = C(n,0)aⁿ b⁰ + C(n,1)a^(n-1) b¹ + C(n,2)a^(n-2) b² + … + C(n,r)a^(n-r) b^r + … + C(n,n)a⁰ bⁿ

where C(n,r) is the binomial coefficient, which is defined as:

C(n,r) = n! / (r! * (n-r)!)

The binomial coefficient C(n,r) gives the number of ways of selecting r items from a set of n items without regard to their order.

The terms in the expansion of (a + b)ⁿ are called binomial terms, and each term is of the form:

C(n,r)a^(n-r) b^r

where r is a non-negative integer and the exponent of a and b sum to n.

Example: Expand (x + y)⁴ using the binomial theorem.

Solution: Using the binomial theorem, we have:

(x + y)⁴ = C(4,0)x⁴ y⁰ + C(4,1)x³ y¹ + C(4,2)x² y² + C(4,3)x¹ y³ + C(4,4)x⁰ y⁴

Simplifying each term using the binomial coefficient formula, we get:

(x + y)⁴ = x⁴ + 4x³ y¹ + 6x² y² + 4x¹ y³ + 1y⁴

Therefore, (x + y)⁴ = x⁴ + 4x³ y + 6x² y² + 4xy³ + y⁴

In conclusion, the binomial theorem is a powerful tool for expanding the power of a binomial expression, and allows us to find the coefficients of each term in the expansion. By understanding and applying this theorem, we can solve various problems involving binomial expressions.

Describe Pascal’s Triangle

Pascal’s Triangle is a triangular arrangement of numbers where the first and last numbers in each row are always 1, and each of the other numbers is the sum of the two numbers directly above it. Pascal’s Triangle is named after Blaise Pascal, a French mathematician who first published the triangle in the 17^th century.

Pascal’s Triangle has many properties and applications in various fields of mathematics, including combinatorics and probability theory. It is also closely related to the binomial theorem, and provides a visual representation of the binomial coefficients.

The first few rows of Pascal’s Triangle are shown below:

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

Each number in Pascal’s Triangle can be found by adding the two numbers directly above it. For example, the number 6 in the fifth row is obtained by adding the numbers 3 and 3 directly above it. Similarly, the number 10 in the sixth row is obtained by adding the numbers 4 and 6 directly above it.

Pascal’s Triangle has several interesting properties. For example, the sum of the numbers in any row of Pascal’s Triangle is equal to 2 raised to the power of the row number. This can be seen by noticing that each row represents the coefficients of the expansion of (1 + 1)ⁿ, where n is the row number.

Pascal’s Triangle is also closely related to the binomial coefficients. The kth number in the nth row of Pascal’s Triangle is equal to the binomial coefficient C(n, k), which gives the number of ways of selecting k items from a set of n items without regard to their order.

Example: Find the 5th number in the 7th row of Pascal’s Triangle.

Solution: The 7th row of Pascal’s Triangle is:

1 6 15 20 15 6 1

To find the 5th number in this row, we look at the 5th position from the left, which is 15. This is the same as the binomial coefficient C(7, 5), which gives the number of ways of selecting 5 items from a set of 7 items without regard to their order. Thus, there are 15 ways of selecting 5 items from a set of 7 items.

In conclusion, Pascal’s Triangle is a useful tool in combinatorics and probability theory, and provides a visual representation of the binomial coefficients. By understanding and applying its properties, we can solve various problems involving combinatorics and probability.

Describe properties of the expansion of (x+a)ⁿ

The expansion of (x + a)ⁿ, also known as the binomial expansion, is a formula for expressing the nth power of a binomial as a sum of terms involving powers of the individual terms. The expansion is given by the binomial theorem, which states that:

(x + a)ⁿ = C(n,0)xⁿ a⁰ + C(n,1)x^(n-1) a¹ + … + C(n,r)x^(n-r) a^r + … + C(n,n)x⁰ aⁿ

where C(n,r) is the binomial coefficient and is equal to n!/(r!(n-r)!).

The binomial expansion has several important properties, some of which are:

1. The coefficients of the expansion are the binomial coefficients C(n,r).
2. The number of terms in the expansion is n+1.
3. The sum of the powers of x and a in each term is n.
4. The expansion is symmetric if a = x or if n is even. This means that the coefficients of the terms in the expansion are the same if x and a are interchanged.
5. The first and last terms of the expansion are both equal to aⁿ and xⁿ, respectively.
6. The sum of the coefficients in the expansion is equal to 2ⁿ.

The properties of the binomial expansion can be used to simplify and evaluate expressions involving binomials. For example, we can use the binomial expansion to find the value of (1.02)¹⁰ as follows:

(1.02)¹⁰ = (1 + 0.02)¹⁰

Using the binomial expansion with n = 10 and a = 0.02, we get:

(1.02)¹⁰ = C(10,0)1¹⁰ 0.02⁰ + C(10,1)1⁹ 0.02¹ + … + C(10,10)1⁰ 0.02¹⁰

= 1 + 0.1(0.02) + 0.45(0.02)² + … + 0.0000000001

= 1.21899444

Thus, (1.02)¹⁰ is approximately equal to 1.21899444.

In conclusion, the binomial expansion of (x+a)ⁿ has several important properties that can be used to simplify and evaluate expressions involving binomials. By understanding and applying these properties, we can solve various problems involving binomials and their expansions.

Find independent term x, middle term, and the r^th term in the expansion of (x+a)ⁿ

The binomial expansion of (x + a)ⁿ is given by the formula:

(x + a)ⁿ = C(n,0)xⁿ a⁰ + C(n,1)x^(n-1) a¹ + … + C(n,r)x^(n-r) a^r + … + C(n,n)x⁰ aⁿ

where C(n,r) is the binomial coefficient and is equal to n!/(r!(n-r)!).

Using this formula, we can find the independent term, middle term, and r^th term in the expansion of (x+a)ⁿ as follows:

1. Independent term:

The independent term is the term in the expansion that contains only the constant term a^r. This occurs when the power of x in the term is zero, that is, when n-r = 0 or r = n. Thus, the independent term in the expansion of (x+a)ⁿ is:

C(n,n)x⁰ aⁿ = aⁿ

1. Middle term:

The middle term is the term in the expansion that has the highest power of x when n is odd, or the two middle terms when n is even. To find the middle term, we need to find the value of r that maximises the power of x in the term C(n,r)x^(n-r) a^r. This occurs when n – 2r = 0 or r = n/2 if n is even. Thus, the middle terms in the expansion of (x+a)ⁿ are:

• If n is odd: The middle term is the term with r = (n-1)/2, given by:

C(n,(n-1)/2)x^((n-1)/2) a(⁽ⁿ⁺¹⁾/2)

• If n is even: There are two middle terms, corresponding to r = n/2-1 and r = n/2. These terms are:

C(n,n/2-1)x^(n/2-1) a^(n/2+1) and C(n,n/2)x^(n/2) a^(n/2)

1. r^th term:

The rth term is the term in the expansion with coefficient C(n,r) and power x^(n-r) a^r. Thus, the rth term in the expansion of (x+a)ⁿ is:

C(n,r)x^(n-r) a^r

For example, let’s find the independent term, middle term, and 5th term in the expansion of (x+3)^6.

1. Independent term: The independent term is the term with r = n = 6, given by:

C(6,6)x⁰ 3⁶ = 729

1. Middle term: The middle terms are the terms with r = 2 and r = 3, given by:

C(6,2)x² 3⁴ = 405 and C(6,3)x³ 3³ = 540

1. 5th term: The 5th term is the term with r = 4, given by:

C(6,4)x² 3⁴ = 540

In conclusion, we can use the binomial expansion formula to find the independent term, middle term, and rth term in the expansion of (x+a)ⁿ. By understanding and applying these concepts, we can solve various problems involving binomials and their expansions.