3 Polynomial equations

When a polynomial expression is equated to zero, a polynomial equation is obtained. Linear and quadratic equations, which you have already met, are particular types of polynomial equation.

Key Point 9

A polynomial equation has the form

a n x n + a n − 1 x n − 1 + … a 2 x 2 + a 1 x + a 0 = 0
where a 0 , a 1 , … , a n are known coefficients, a n ≠ 0 , and x represents an unknown whose value(s) are to be found.

Polynomial equations of low degree have special names. A polynomial equation of degree 1 is a linear equation and such equations have been solved in Section 3.1. Degree 2 polynomials are called quadratics; degree 3 polynomials are called cubics; degree 4 equations are called quartics and so on. The following are examples of polynomial equations:

5 x 6 − 3 x 4 + x 2 + 7 = 0 , − 7 x 4 + x 2 + 9 = 0 , t 3 − t + 5 = 0 , w 7 − 3 w − 1 = 0

Recall that the degree of the equation is the highest power of x occurring. The solutions or roots of the equation are those values of x which satisfy the equation.

Key Point 10

A polynomial equation of degree n has n roots.

Some (possibly all) of the roots may be repeated.

Some (possibly all) of the roots may be complex.

Example 24

Verify that x = − 1 , x = 1 and x = 0 are solutions (roots) of the equation

x 3 − x = 0

Solution

We substitute each value in turn into x 3 − x .

( − 1 ) 3 − ( − 1 ) = − 1 + 1 = 0

so x = − 1 is clearly a root.

It is easy to verify similarly that x = 1 and x = 0 are also solutions.

In the next subsection we will consider ways in which polynomial equations of higher degree than quadratic can be solved.

Exercises

Verify that the given values are solutions of the given equations.

  1. x 2 − 5 x + 6 = 0 , x = 3 , x = 2
  2. 2 t 3 + t 2 − t = 0 , t = 0 , t = − 1 , t = 1 2 .