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Warm up

Warm up. Use the Rational Root Theorem to determine the Roots of : x³ – 5x² + 8x – 6 = 0. Lesson 4-5 Fundamental Theorem of Algebra. Objective: To learn & apply the fundamental theorem of algebra & the linear factor theorem.

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Warm up

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  1. Warm up • Use the Rational Root Theorem to determine the Roots of : x³ – 5x² + 8x – 6 = 0

  2. Lesson 4-5 Fundamental Theorem of Algebra Objective: To learn & apply the fundamental theorem of algebra & the linear factor theorem.

  3. We have seen that if a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. This result is called the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra If f (x) is a polynomial of degree n, where n  1, then the equation f (x) = 0 has at least one complex root.

  4. The Linear Factor Theorem Just as an nth-degree polynomial equation has n roots, an nth-degree polynomial has n linear factors. This is formally stated as the Linear Factor Theorem. The Linear Factor Theorem If f (x) =anxn+ an-1xn-1+ … + a1x + a0 b, where n  1 and an 0 , then f (x) = an (x - c1) (x - c2) … (x -cn) where c1, c2,…, cn are complex numbers (possibly real and not necessarily distinct). In words: An nth-degree polynomial can be expressed as the product of n linear factors.

  5. This is the linear factorization for a fourth-degree polynomial. f(x) = an(x - c1)(x - c2)(x - c3)(x - c4) Use the given zeros: c1= -2, c2= 2, c3= i, and, from above, c4= -i. = an(x + 2)(x -2)(x - i)(x + i) = an(x2- 4)(x2+ 1) Multiply f(x) = an(x4- 3x2- 4) Complete the multiplication more more 3.5: More on Zeros of Polynomial Functions EXAMPLE: Finding a Polynomial Function with Given Zeros Find a fourth-degree polynomial function f(x) with real coefficients that has -2, 2, and i as zeros and such that f(3) = -150. Solution Because iis a zero and the polynomial has real coefficients, the conjugate must also be a zero. We can now use the Linear Factorization Theorem.

  6. EXAMPLE: Finding a Polynomial Function with Given Zeros Find a fourth-degree polynomial function f(x) with real coefficients that has -2, 2, and i as zeros and such that f(3) = -150. Solution f (3) = an(34 - 3 • 32- 4) = -150To find an, use the fact that f (3) = -150. an(81 - 27 - 4) = -150Solve for an. 50an= -150 an= -3 Substituting -3 for an in the formula for f(x), we obtain f(x) = -3(x4- 3x2- 4). Equivalently, f(x) = -3x4+ 9x2+ 12.

  7. Multiplicity • Multiplicity refers to the number of times that root shows up as a factor • Ex: if -2 is a root with a multiplicity of 2 then it means that there are 2 factors :(x+2)(x+2)

  8. Practice • Find the polynomial that has the indicated zeros and no others: • -3 of multiplicity 2, 1 of multiplicity 3 • Find the polynomial P(x) of lowest degree that has the indicated zeros and satisfies the given condition: • 2 + 3i and 4 are roots, f(3) = -20 • Answer: f(x) = -16x2 + 58x - 104

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