Solve Polynomial Equations

What you'll learn

Solve polynomial equations by factoring
Use the zero product property to find all roots
Relate the roots to the x-intercepts of the graph

Words to know

Polynomial equation —: an equation set equal to 0 with polynomial terms
Root —: a value that makes the polynomial equal 0
Zero product property —: if a product is 0, a factor is 0
Degree —: the highest exponent in the polynomial
x-intercept —: where the graph crosses the x-axis

Why it matters Polynomial equations model volume, motion, and growth, and their roots tell us where those quantities reach a target value.

Worked examples

Model ItI do

Solve (x + 2)(x - 1)(x - 3) = 0.

By the zero product property, set each factor to 0: x + 2 = 0, x - 1 = 0, or x - 3 = 0. The roots are x = -2, x = 1, and x = 3. The graph crosses the x-axis at exactly these three points.

Try It TogetherWe do

Solve x^2(x - 4) = 0.

Set each factor to 0: x^2 = 0 gives x = 0, and x - 4 = 0 gives x = 4. The roots are x = 0 (a double root) and x = 4.

Now YouYou do

Solve x^3 - 4x = 0.

Factor out x, then factor the remaining difference of squares. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

1.Solve x(x - 8) = 0.
2.Solve (x + 2)(x - 7) = 0.
3.Solve x^2 - 25 = 0.

On Level

The core Quick Check from the book.

1.Solve x(x - 5) = 0.
2.Solve (x + 1)(x - 6) = 0.
3.Solve x^2 - 9 = 0.
4.How many roots does a degree-3 polynomial have at most?
5.Solve x^3 = 0.

Challenge

Stretch your thinking.

1.Solve x^3 - 9x = 0.
2.Solve x^3 + 2x^2 - 3x = 0.
3.Solve x^4 - 13x^2 + 36 = 0.