### 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) (4*x* + 1) (4 *x* + 3) = 0.

Set each factor equal to zero. This gives a set of equations:
*x* − 1 = 0

4*x* + 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}.

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