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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? 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.) Find the number of solutions or zeros 1. How many solutions does the equation x4 + 5x2 – 36 = 0 have? ANSWER 4 2. How many zeros does the function f (x) = x3 + 7x2 + 8x – 16 have? ANSWER 3 • 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. Solve the Polynomial Equation. x3 + x2 −x − 1 = 0 1 1 1 1 1 2 1 1 −1 −1 2 1 1 0 Notice that −1 is a solution two times. This is called a repeated solution, repeated zero, or a double root. x2 + 2x + 1 (x + 1)(x + 1) x = −1, x = −1, x = 1 http://my.hrw.com/math06_07/nsmedia/t ools/Graph_Calculator/graphCalc.html Finding the Number of Solutions or Zeros x x3 + 3x2 + 16x + 48 = 0 x2 (x + 3)(x2 + 16)= 0 x + 3 = 0, x2 + 16 = 0 x = −3, x2 = −16 +16 x = − 3, x = ± 4i +3 x3 3x2 16x 48 Find the zeros of a polynomial equation Find all zeros of f (x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14. SOLUTION STEP 1 Find the rational zeros of f. Because f is 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: f (x) = (x + 1)2(x – 2)(x2 – 4x + 7) 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 , and 2 – i 3. Finding the Number of Solutions or Zeros Zeros: −2,−2,−2, 0 Finding the Zeros of a Polynomial Function Find all the zeros of f(x) = x5 − 2x4 + 8x2 − 13x + 6 Possible rational zeros: ±6, ±3, ±2, ±1 1 −2 1 1 −2 1 0 8 −13 6 1 −1 −1 −1 −1 7 −2 6 −10 7 −6 6 −6 0 1 −3 5 −3 1 −2 3 1 −2 3 0 x2 −2x + 3 0 1,1, 2,1 i 2 ,1 i 2 Use quadratic formula or complete the square • 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. Assignment is: Page 141, 3-9 all, 11-17 odd Show your work NO WORK NO CREDIT