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

2. You have used several methods for factoring polynomials
You have used several methods for factoring polynomials. As with some quadratic equations, factoring a polynomial equation is one way to find its real roots.

3. Do Now:
Solve the polynomial equation by factoring.
x3 – 2x2 – 25x = –50
2x6 – 10x5 – 12x4 = 0

4. Try This!

5. 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.

6. You cannot always determine the multiplicity of a root from a graph
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.

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

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

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

10. Do Now:
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.

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

12. Check It Out! Example 3 Continued
Step 4 Factor the polynomial.

13. Polynomial equations may also have irrational roots.

14. The Irrational Root Theorem say that irrational roots come in conjugate pairs. For example, if you know that 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.

15. Do Now
Identify all the real roots of 2x3 – 3x2 –10x – 4 = 0.
Step 2 Graph y = 2x3 – 3x2 –10x – 4 to find the x-intercepts.

16. Check It Out! Example 4 Continued1 2
Step 3 Test the possible rational root – .
Step 4 Solve 2x2 – 4x – 8 = 0 to find the remaining roots.

17. Notice that the degree of the function in Example 1 is the same as the number of zeros. This is true for all polynomial functions. However, all of the zeros are not necessarily real zeros. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers.

18. Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots.
The polynomial is of degree 4, so there are exactly four roots for the equation.
Step 1 Use the rational Root Theorem to identify rational roots.
Step 2 Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots.

19. Step 3 Test the possible real roots.
Step 4 Find the remaining roots.

20. Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all roots.
The polynomial is of degree 4, so there are exactly four roots for the equation.
Step 1 Use the rational Root Theorem to identify rational roots.
Step 2 Graph y = x4 + 4x3 – x2 + 16x – 20 to find the real roots.

21. Step 3 Test the possible real roots.
Step 4 Find the remaining roots.

22. You have learned several important properties about real roots of polynomial equations.
You can use this information to write polynomial function when given in zeros.

23. Writing Polynomial Functions2 3
Write the simplest polynomial with roots –1, , and 4.
Step 1 Identify all roots.
Step 2 Write the equation in factored form.

24. Step 3 Multiply.

25. Do Now:
Write the simplest polynomial function with the given zeros.
–2, 2, 4

27. Write the simplest function with zeros 2i, , and 3.
1+ 2
Step 1 Identify all roots.
Step 2 Write the equation in factored form.

28. Step 3 Multiply.

29. Try This!
Write the simplest function with zeros 1+i, -3, and .