10.3 - Polynomial equations
Before reading this section you may want to review the following topics:
To solve a polynomial equation, follow these steps:
- Ensure that the polynomial equation is in standard form,
which is to have a polynomial on the left-hand-side of the equation and zero on the right-hand-side.
- Factor the polynomial on the left-hand-side
completely using the methods of section 10.2.
- In section 4.1, we saw that if
a · b = 0 then a = 0 or b = 0.
Use this fact to set each factor equal to zero. This replaces a polynomial equation
with a set of new equations that are either linear or quadratic.
- Solve each of the new equations. The solution set for the polynomial equation
is the collection of all the solutions of the new equations.
Example: Solve the polynomial equation
16 x 3 − 13 x = 3.
Solution: First put the polynomial equation into standard form:
16 x 3 − 13 x − 3 = 0.
Factor the left-hand-side using the deflation method:
(x − 1) (4x + 1) (4 x + 3) = 0.
Set each factor equal to zero. This gives a set of equations:
x − 1 = 0
4x + 1 = 0
4 x + 3 = 0
Solve them. This gives a set of solutions:
x = 1
x = −¼
x = −¾
The solution set is x = {1, −¼, −¾}.
If you found this page in a web search you won’t see the
Table of Contents in the frame on the left.
Click here to display it.