7.4 Solving Polynomial 7.4 Solving Polynomial EquationsEquations

7.4 Solving Polynomial 7.4 Solving Polynomial EquationsEquations

Objectives: •Solve polynomial equations•Find the real zeros of polynomial functions and state the multiplicity of each

Example 1Use factoring to solve 5x3 – 12x2 + 4x = 0. x(5x2 – 12x + 4) =

0x(5x – 2)(x – 2) = 0

x = 0 2

, x5

, x = 2

Example 2Use a graph, synthetic division, and factoring to find all roots of x3 + 3x2 – 4 = 0.

First, graph the polynomial function to approximate the roots.

Then use synthetic division to test your choices.

1

1

4

4

4

4

0

1 1 3 0 -4

Since the remainder is 0, x – 1 is a factor of x3 + 3x2 - 4.

Example 2Use a graph, synthetic division, and factoring to find all roots of x3 + 3x2 – 4 = 0.

Since the remainder is 0, x – 1 is a factor of x3 + 3x2 - 4.

x3 + 3x2 – 4 = 0(x – 1)(x2 + 4x + 4) = 0(x – 1)(x + 2)(x + 2) = 0x = 1

x = -2 x = -2

The roots of x3 + 3x2 – 4 are 1 and -2, with the root -2 occurring twice.

PracticeUse a graph, synthetic division, and factoring to find all of the roots of x3 + 2x2 – 4x – 8 = 0.

Example 3Use variable substitution and factoring to find all roots of x4 – 5x2 – 6 = 0.

Substitute u for x2 in the above equation and then solve for u.

(x2)2 – 5(x2) – 6 = 0 u2 – 5u – 6 = 0

(u – 6)(u + 1) = 0u = 6 u = -

1x2 = 6

x2 = -1x 6 x 1

PracticeUse variable substitution and factoring to find all of the roots of x4 – 9x2 + 14 = 0.

Example 4Find the real zeros of the function. Give approximate values to the nearest hundredth, if necessary.

f(x) = x4 – 10x3 + 22x2 + 20x - 48

x = 4 x = 6

x = 1.41

x = -1.41

Homework

Lesson 7.4 Exercises 11-25 odd and 39-49 odd