1. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Intelligence is knowing that a tomato is a fruit; Wisdom is not putting it in a fruit salad.

2. Warm Up Identify all the real roots of each equation. 0, –2, 4 1. 4x 5 – 8x 4 – 32x 3 = 0 2. x 3 –x 2 + 9 = 9x 1, –3, 3

3. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra The Following are Equivalent A real number r is a root of the polynomial equation P(x) = 0 P(r) = 0 r is an x-intercept of the graph of P(x) (x – r) is a factor of P(x) When you divide the rule for P(x) by x – r, the remainder is 0 r is a zero of P(x) r is a solution of P(x)

4. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Zeros of a Polynomial Function A Polynomial Function is usually written in function notation or in terms of x and y. The Zeros of a Polynomial Function are the solutions to the equation you get when you set the polynomial equal to zero.

5. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Graph of a Polynomial Function

6. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Steps to Solve (find the roots of) a Polynomial Function 1.Graph to find initial root/roots 2.Use Synthetic Division to verify root(s) and factor. 3.Use quadratic formula or continue factoring to identify remaining roots.

7. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Example: Solve the Polynomial Equation

8. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Practice: Solve the Polynomial Equation

9. The degree of the function is the same as the number of zeros. However, all of the zeros are not necessarily real zeros. Polynomial functions, like quadratic functions, may have complex zeros that are not real numbers.

10. Example 1: Finding All Roots of a Polynomial Solve x 4 – 3x 3 + 5x 2 – 27x – 36 = 0 by finding all roots. The polynomial is of degree 4, so there are exactly four roots for the equation. Find the real roots at or near –1 and 4. Graph to find the real roots.

11. Example 1 Continued Test –1 The remainder is 0, so (x + 1) is a factor. Test the possible real roots. 1 –3 5 –27 –36–1–1 –1–1 1 –49–360 4–936 Test 4 in the cubic polynomial. The remainder is 0, so (x – 4) is a factor. 1 –4 9 –36 4 4 1 The polynomial factors into (x + 1)(x 3 – 4x 2 + 9x – 36) = 0. 360 0 09 The polynomial factors into (x + 1)(x – 4)(x 2 + 9) = 0.

12. Example 1 Continued Step 4 Solve x 2 + 9 = 0 to find the remaining roots. The polynomial factors into (x + 1)(x – 4)(x 2 + 9) = 0. x 2 + 9 = 0 x 2 = –9 The fully factored form of the equation is (x + 1)(x – 4)(x + 3i)(x – 3i) = 0. The solutions are 4, –1, 3i, –3i.

13. Solve x 4 + 4x 3 – x 2 +16x – 20 = 0 by finding all roots. The polynomial is of degree 4, so there are exactly four roots for the equation. Check It Out! Example 2 Find the real roots at or near –5 and 1. Test –5. (x + 5) is a factor. 1 4 –1 16 –20–5–5 –5–5 1 –14–40 5–2020 Test 1. (x + 5)(x 3 – x 2 + 4x – 4) = 0 1 –1 4 –4 1 1 1 040 04 (x -1) is a factor.

14. The polynomial factors into (x + 5)(x – 1)(x 2 + 4) = 0. The fully factored form of the equation is (x + 5) (x – 1)(x + 2i)(x – 2i) = 0. The solutions are –5, 1, –2i, +2i. Check It Out! Example 2 Continued

15. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Writing a Polynomial given the roots.

16. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Example 1: The zeros of a third-degree polynomial are 2 (multiplicity 2) and -5. Write a polynomial. (x – 2)(x – 2)(x – (-5)) = (x – 2)(x – 2)(x+5) First, write the zeros in factored form Second, multiply the factors out to find your polynomial

17. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Example 1 Continued (x – 2)(x – 2)( x+5 ) First, FOIL or box two of the factors Second, box your answer from above with your remaining factors to get your polynomial:

18. Example 2: Writing Polynomial Functions

19. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra More Examples: Given the following zeros, write the lowest degree polynomial

20. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra More Examples: Given the following zeros, write the lowest degree polynomial

21. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra More Examples: Given the following zeros, write the lowest degree polynomial

22. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra More Examples:

23. Holt McDougal Algebra 2 3-6 Fundamental Theorem of Algebra Homework 3.6: p 193:11-19 (odd) 20-22, 27-33 (odd) 36