Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

Identify the multiplicity of roots.Use the Rational Root Theorem and the irrational Root Theorem to solve polynomial equations.

Objectives

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

In Lesson 6-4, you used several methods for factoring polynomials. As with some quadratic equations, factoring a polynomial equation is one way to find its real roots.

Recall the Zero Product Property from Lesson 5-3. 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

6-5Finding Real Roots of Polynomial Equations

Solve the polynomial equation by factoring.

Example 1A: Using Factoring to Solve Polynomial Equations

4x6 + 4x5 – 24x4 = 0

Factor out the GCF, 4x4.4x4(x2 + x – 6) = 0

Factor the quadratic.4x4(x + 3)(x – 2) = 0

Set each factor equal to 0.

4x4 = 0 or (x + 3) = 0 or (x – 2) = 0

Solve for x.x = 0, x = –3, x = 2

The roots are 0, –3, and 2.

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

Example 1B: Using Factoring to Solve Polynomial Equations

Solve the polynomial equation by factoring.

x4 + 25 = 26x2

Set the equation equal to 0.x4 – 26 x2 + 25 = 0

Factor the trinomial in quadratic form.

(x2 – 25)(x2 – 1) = 0

Factor the difference of two squares.

(x – 5)(x + 5)(x – 1)(x + 1)

Solve for x.

x – 5 = 0, x + 5 = 0, x – 1 = 0, or x + 1 =0

The roots are 5, –5, 1, and –1.

x = 5, x = –5, x = 1 or x = –1

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

x3 – 2x2 – 25x = –50

Set the equation equal to 0.x3 – 2x2 – 25x + 50 = 0

Factor.(x + 5)(x – 2)(x – 5) = 0

Solve for x.

x + 5 = 0, x – 2 = 0, or x – 5 = 0

The roots are –5, 2, and 5.

x = –5, x = 2, or x = 5

Solve the polynomial equation by factoring.

Check It Out! Example 1b

Holt McDougal Algebra 2

6-5Finding 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

6-5Finding Real Roots of Polynomial Equations

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

Example 2A: Identifying Multiplicity

x3 + 6x2 + 12x + 8 = (x + 2)(x + 2)(x + 2)

x + 2 is a factor three times. The root –2 has a multiplicity of 3.

x3 + 6x2 + 12x + 8 = 0

Check Use a graph. A calculator graph shows a bend near (–2, 0).

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

Check It Out! Example 2a

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

x4 – 8x3 + 24x2 – 32x + 16 = (x – 2)(x – 2)(x – 2)(x – 2)

x – 2 is a factor four times. The root 2 has a multiplicity of 4.

x4 – 8x3 + 24x2 – 32x + 16 = 0

Check Use a graph. A calculator graph shows a bend near (2, 0).

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

Not all polynomials are factorable, but the Rational Root Theorem can help you find all possible rational roots of a polynomial equation.

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

Polynomial equations may also have irrational roots.

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

Example 4: Identifying All of the Real Roots of a Polynomial Equation

Identify all the real roots of 2x3 – 9x2 + 2 = 0.

Step 1 Use the Rational Root Theorem to identify possible rational roots.

±1, ±2 ±1, ±2 = ±1, ±2, ± .1

2 p = 2 and q = 2

Step 2 Graph y = 2x3 – 9x2 + 2 to find the x-intercepts.

The x-intercepts are located at or near –0.45, 0.5, and 4.45. The x-intercepts –0.45 and 4.45 do not correspond to any of the possible rational roots.

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

2 –9 0 2

1

2 0–2–4

–4–8

Step 3 Test the possible rational root .

Example 4 Continued

1 2

1 2

Test . The remainder is

0, so (x – ) is a factor.1 2

1 2

The polynomial factors into (x – )(2x2 – 8x – 4).1 2

Step 4 Solve 2x2 – 8x – 4 = 0 to find the remaining roots.2(x2 – 4x – 2) = 0 Factor out the GCF, 2

Use the quadratic formula to identify the irrational roots.

4± 16+8 2 62

x

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

Example 4 Continued

The fully factored equation is

12 x – x – 2 + 6 x – 2 – 6 = 0

2

The roots are , , and .1 2

2 6 2 6

Holt McDougal Algebra 2

6-5Finding Real Roots of Polynomial Equations

4. A box is 2 inches longer than its height. The width is 2 inches less than the height. The volume of the box is 15 cubic inches. How tall is the box?

Lesson Quiz

2. 5x4 – 20x3 + 20x2 = 0

1. x3 + 9 = x2 + 9x

4 with multiplicity 33. x3 – 12x2 + 48x – 64 = 0

–3, 3, 1

0 and 2 each with multiplicity 2

5. Identify all the real roots of x3 + 5x2 – 3x – 3 = 0.

3 in.

Solve by factoring.

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

1, 3 + 6, 3 6