5C - Methods for solving polynomial equations

Solving quadratic equations

In section 4 we examined solving quadratic equations (which are among the simplest forms of polynomial equations) by factorising and using the null factor law. In this section we will extend this method to solving cubic, quartic and higher order polynomials equations using similar techniques.

When we solved quadratic equations we started by using inverse operations to make one side of the equation equal zero. We then factorised the expression on the other side of the equation to find the linear factors and finally we solved using the null factor law.

Factorising polynomials

In order to solve polynomial equations we need to be able to factorise them. For cubic, quartic and higher order polynomials this is not as straight forward as quadratics where we could often inspect the trinomial to determine the linear factors. For polynomials we need to first identify factors using the factor theorem and then divide the polynomial using either:

• 5C.1 - Long division
• 5C.2 - Synthetic division

Solving polynomial equations

Recap: The null factor law

After the polynomial has been divided, it can be expressed as a product of its factors. Once this is the case we can use the null factor law (stated below) to solve the equation.

Method for solving polynomial equations

To solve polynomial equations, use the following steps:

1. Get the polynomial equation to equal zero. For example: ax³ + bx² + cx + d = 0.
2. Identify factors of the polynomial expression using the factor theorem.
1. Divide the polynomial expression using long division or synthetic division.
2. Express the polynomial as a product of its' linear factors.
   1. Use the null factor law to find the solution(s).

Before viewing the following examples it is recommended to learn how to perform long division and synthetic division. The examples below will primarily demonstrate synthetic division; however, it is possible to obtain the same result using other methods of division.

5C - VIDEO EXAMPLE 1:

Solve the following polynomial equation for x:

5C - VIDEO EXAMPLE 2:

Solve the following polynomial equation for x:

5C - VIDEO EXAMPLE 3:

Solve the following polynomial equation for x:

5C - VIDEO EXAMPLE 4:

Solve the following polynomial equation for x:

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