Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

• How do we identify the multiplicity of roots?

• How do we use the Rational Root Theorem and the irrational Root Theorem to solve polynomial equations?

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

In Lesson 3-5, you used several methods for factoring polynomials. As with some quadratic equations, factoring a polynomial equation is one way to find its real roots.

Using the Zero Product Property, you can find the roots, or solutions, of the polynomial equation P(x) = 0 by setting each factor equal to 0 and solving for x.

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

Solve the polynomial equation by factoring.

Example 1: Using Factoring to Solve Polynomial Equations

4x6 + 4x5 – 24x4 = 0 44x 2x 6x

x x3 2 0 4 4 x

0

04 4 x 03 x 02 x0x 2x3x

Check using a graph.

The roots appear to be located at x = 0, x = –3, and x = 2.

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

Solve the polynomial equation by factoring.

Example 2: Using Factoring to Solve Polynomial Equations

x4 + 25 = 26x2

02526 24 xx

0 2x2x 25 1

0 1x1x5x5x

01 x 01 x05 x05 x5x 5x 1x 1x

The roots are 5, –5, 1, and –1.

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

0

Solve the polynomial equation by factoring.

Example 3: Using Factoring to Solve Polynomial Equations

2x6 – 10x5 – 12x4 = 0 2x 642x x5

02 4 x 06 x 01 x0x 6x 1x

0 2 4 x x x 16

The roots are 0, 6, and –1.

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

02 x

0

Solve the polynomial equation by factoring.

Example 4: Using Factoring to Solve Polynomial Equations

x3 – 2x2 – 25x = –50

050252 23 xxx

The roots are 5, –5, and 2.

2 25 22x

252 x 02 x

5x5x02 x05 x05 x

2x5x5x

x x

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root. In 3x5 + 18x4 + 27x3 = 0 has two multiple roots, 0 and –3. For example, the root 0 is a factor three times because 3x3 = 0.

The multiplicity of root r is the number of times that x – r is a factor of P(x). When a real root has even multiplicity, the graph of y = P(x) touches the x-axis but does not cross it. When a real root has odd multiplicity greater than 1, the graph “bends” as it crosses the x-axis.

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

You cannot always determine the multiplicity of a root from a graph. It is easiest to determine multiplicity when the polynomial is in factored form.

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

0 2 2 x

Identify the roots of each equation. State the multiplicity of each root.

Example 5: Identifying Multiplicity

2x4 12x3 + 18x2 = 0

0 2x

0x

x x

The root 0 has multiplicity of 2, the root 3 has multiplicity of 2.

22x x6 933

03 x03 x02 2 x3x3x

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

x3 – x2 – x + 1 = 0

Identify the roots of each equation. State the multiplicity of each root.

Example 6: Identifying Multiplicity

0 2x 1 1 1 01 x 12 x

1x1x01 x01 x01 x

1x1x1x

01 x

The root 1 has multiplicity 2, the root 1 has multiplicity 1.

x x

Holt McDougal Algebra 2

Finding Real Roots of Polynomial Equations

Lesson 4.1 Practice A