Identify the multiplicity of roots.

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

2. Vocabulary multiplicity

3. 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.

4. Check It Out! Example 1a Solve the polynomial equation by factoring. 2x6 – 10x5 – 12x4 = 0

5. Check It Out! Example 1b Solve the polynomial equation by factoring. x3 – 2x2 – 25x = –50

6. 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.

7. 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.

8. Check It Out! Example 2a Identify the roots of each equation. State the multiplicity of each root. x4 – 8x3 + 24x2 – 32x + 16 = 0

9. Check It Out! Example 2b Identify the roots of each equation. State the multiplicity of each root. 2x6 – 22x5 + 48x4 + 72x3 = 0

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

11. Check It Out! Example 3 A shipping crate must hold 12 cubic feet. The width should be 1 foot less than the length, and the height should be 4 feet greater than the length. What should the length of the crate be? Step 1 Write an equation to model the volume of the box. Let x represent the length in feet. Then the width is x – 1, and the height is x + 4. V = lwh. Multiply the left side. Set the equation equal to 0.

12. p q Check It Out! Example 3 Continued Step 2Use the Rational Root Theorem to identify all possible rational roots. Step 3Test the possible roots to find one that is actually a root. The width must be positive, so try only positive rational roots.

13. Check It Out! Example 3 Continued Step 4 Factor the polynomial. The synthetic substitution of 2 results in a remainder of 0, so 2 is a root and the polynomial in factored form is (x – 2)(x2 + 5x + 6).

14. Polynomial equations may also have irrational roots.

15. The Irrational Root Theorem say that irrational roots come in conjugate pairs. For example, if you know that 1 + is a root of x3 – x2 – 3x – 1 = 0, then you know that 1 – is also a root. Recall that the real numbers are made up of the rational and irrational numbers. You can use the Rational Root Theorem and the Irrational Root Theorem together to find all of the real roots of P(x) = 0.

16. Check It Out! Example 4 Identify all the real roots of 2x3 – 3x2 –10x – 4 = 0. p = –4 and q = 2 Step 2Graph y = 2x3 – 3x2 –10x – 4to find the x-intercepts.

17. 1 1 2 2 Check It Out! Example 4 Continued Step 3 Test the possible rational root – . 2 –3 –10 –4 Step 4 Solve 2x2 – 4x – 8 = 0 to find the remaining roots.

18. ( ) ( ) æ ö 1 ù é é ù 2 x + x – 1+ 5 x – 1– 5 ç ÷ û ë ë û 2 è ø The roots are – , , and . 1 5 + 1 5 - 1 2 Check It Out! Example 4 Continued The fully factored equation is