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Factoring Polynomials in Algebraic Equations

By Kathleen Knowles, 03 Jul 2020

A polynomial is a mathematical expression containing variables in which the operators in the expression are limited to addition, subtraction, and multiplication; and non-negative integer powers of the variables. The variables are referred to as indeterminates since the values of the variables are unknown. Numerical values attached to the variables are called coefficients. An example of a polynomial is 2x + 6. In this polynomial, the variable is x, the coefficient of x is 2. The number 6 in the polynomial is a constant. The above polynomial is a polynomial in a single indeterminate x. Polynomials could be in multiple indeterminates such as 2yx2 – 6x + 21. The indeterminates in the polynomial are x and y.

Polynomials also have another property called degree. The degree of a term of a polynomial is the sum of the exponents of the variables of that term. In the polynomial 2yx2 – 6x + 21, the degree of the term 2yx2 is 3 because the exponent of y is 1 and the exponent of x2 is 2. The sum of the exponents is 3. The degree of a polynomial is the degree of the term with the highest degree. In the polynomial 2yx2 – 6x + 21, the term 2xy has a degree of 3, 6x has degree of 1 and 21 is a constant so has a degree of 0. Since the highest degree of the terms is 3, the degree of the polynomial is 3.

Some algebraic equations involve polynomials. This is usually one polynomial being equated to another polynomial. An example is 2xy – 3 = 4x2. Both sides of the equation are polynomials of degree 2.

Factoring Polynomials

One of the methods of simplifying polynomials is by factorization. Factorization is the process of rewriting the polynomial as a multiple of the common factors of the terms of the polynomial. Factorization relies on the distributive law which states that

a(b + c) = ab + bc.

To understand how to perform factorization, one needs to understand the common highest factor (HCF). The HCF is the largest number that can go into all the terms of the polynomial without a remainder.

Example

What is the HCF of the polynomial 2x3y – 6x2

The highest common factor of the polynomial is 2x2 because dividing all the terms by 2x2 gives

2x3y / 2x2 = xy and

6x2 / 2x2 = 3

2x2 is the largest number that both terms of the polynomial can divide without a remainder.

Linear Polynomials

Linear polynomials are polynomials of degree one. Polynomials of the first degree can be factorised as shown in the example below.

Example

Factorize the polynomial algebraic equation 2x – 6 = x and hence find x.

Step 1

Collect like terms are put them on one side of the equation. Note that when a term crosses the equality sign, the sign of the term changes. If it was positive, it changes to negative and vice versa.

2x – 6 = x

2x – x = 6

Step 2

Factorize the side of the equation with common factors. For this equation, the side with common factors is the left-hand side.

2x – x = 6

x(2 – 1) = 6

Step 3

Solve the part in the bracket

x = 6

The solution to the equation 2x – 6 = x is 6. The normal rules of manipulating algebra can be used to factorize and solve linear polynomials.

Quadratic Polynomials

Polynomials with a degree of 2 are called quadratic polynomials. When factorizing quadratic polynomials, it is important to first rewrite the quadratic equation such that there is no parenthesis, and all like terms are grouped on one side of the equation. An example will illustrate the steps to factorize a quadratic equation.

Example 1

Solve the quadratic equation using factorization 2x2 = 3x

Step 1

Rewrite the equation such that all the like terms are on one side of the equation

2x2 – 3x = 0

Step 2

Factor out the common factor on the right-hand side of the equation.

x(2x – 3) = 0

Step 3

Equate each of the factors to zero and solve.

x = 0;

2x – 3 = 0

Take 3 to the right-hand side of the equation

2x = 3

x = 3/2

the solutions to the quadratic equation are x = 0 and x = 3/2

Example 2

Solve the quadratic equation using factorization 3x2 – 10 = x

Step 1

Rewrite the equation and group like terms together. Also, set the equation to zero.

3x2 – 10 = x

3x2 – x – 10 = 0

x has crossed the equal to sign and it has changed to negative

Step 2

Find the factors of the right-hand side of the equation. To find the factors of this special quadratic equation in which there is a constant, use the following short cut:

  1. Multiply the coefficient of x2 by the constant term. 3 * (-10) = -30
  2. Find two numbers such that when they are added, the answer is the coefficient of x, and when the two numbers are multiplied, the answer is the product obtained in 1 above.

The coefficient of x is -1

The two numbers are 5 and -6

5 – 6 = -1

5 * (-6) = -30

  1. Substitute the x-term with the two numbers such that the answer is the x-term
  2. Find the factors of the first two terms and the factors of the last two terms.

x(3x + 5) – 2(3x + 5) = 0

  1. Factorize out the common term in parenthesis

(3x + 5)(x – 2) = 0

Step 3

Equate each of the factors to zero and solve for x

3x – 5 = 0

3x = 5

x = 5/3

And

x – 2 = 0

x = 2

The solutions to the quadratic equation are x = 5/3 and x = 2.

It's important to note that polynomials with no common factors cannot be factorized.

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