2.7 Using the Fundamental Theorem of Algebra

1. 2.7 Using the Fundamental Theorem of Algebra What is the fundamental theorem of Algebra? What methods do you use to find the zeros of a polynomial function?

2. Find the number of solutions or zeros a. How many solutions does the equation x3 + 5x2 + 4x + 20 = 0 have? SOLUTION Because x3 + 5x2 + 4x + 20 = 0 is a polynomial equation of degree 3,it has three solutions. (The solutions are – 5, – 2i, and 2i.)

3. ANSWER ANSWER 4 3 Find the number of solutions or zeros 1. How many solutions does the equation x4 + 5x2 – 36 = 0 have? 2. How many zeros does the function f (x) = x3 + 7x2 + 8x – 16 have?

4. German mathematician Carl Friedrich Gauss (1777-1855) first proved this theorem. It is the Fundamental Theorem of Algebra. If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers. If f(x is a polynomial of degree n where n>0, then the equation f(x) has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions, and so on.

5. Solve the Polynomial Equation. x3 + x2 −x − 1 = 0 Notice that −1 is a solution two times. This is called a repeated solution, repeated zero, or a double root. • 1 −1 −1 1 1 2 1 1 2 1 0 x2 + 2x + 1 (x + 1)(x + 1) x = −1, x = −1, x = 1 http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

6. Finding the Number of Solutions or Zeros x +3 x3 + 3x2 + 16x + 48 = 0 (x + 3)(x2 + 16)= 0 x + 3 = 0, x2 + 16 = 0 x = −3, x2 = −16 x = − 3, x = ± 4i x2 x3 3x2 16x 48 +16

7. STEP 1 Find the rational zeros of f. Because fis a polynomial function of degree 5, it has 5 zeros. The possible rational zeros are + 1, + 2, + 7, and +14. Using synthetic division, you can determine that – 1 is a zero repeated twice and 2 is also a zero. STEP 2 Write f (x) in factored form. Dividing f (x) by its known factors x + 1, x + 1, and x – 2 gives a quotient of x2 – 4x + 7. Therefore: Find the zeros of a polynomial equation Find all zeros of f (x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14. SOLUTION f (x) = (x + 1)2(x – 2)(x2 – 4x + 7)

8. STEP 3 Find the complex zeros of f . Use the quadratic formula to factor the trinomial into linear factors. f(x) = (x + 1)2(x – 2) x – (2 + i 3 ) x – (2 – i 3 ) ANSWER The zeros of f are – 1, – 1, 2, 2 + i 3 , and2 – i 3.

9. Finding the Number of Solutions or Zeros Zeros: −2,−2,−2, 0

10. Finding the Zeros of a Polynomial Function Possible rational zeros: ±6, ±3, ±2, ±1 1 −2 0 8 −13 6 Find all the zeros of f(x) = x5− 2x4 + 8x2 − 13x + 6 1 1 −1 −1 7 −6 1 −1 −1 7 −6 0 −2 −2 6 −10 6 1 −3 5 −3 0 1 1 −2 3 1 −2 3 0 x2−2x + 3 Use quadratic formula or complete the square

11. What is the fundamental theorem of Algebra? If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers. What methods do you use to find the zeros of a polynomial function? Rational zero theorem (2.6) and synthetic division.

12. Assignment is: Page 141, 3-9 all, 11-17 odd Show your work NO WORK NO CREDIT