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

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Objective: To learn & apply the fundamental theorem of algebra & the linear factor theorem.

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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 If f (x) is a polynomial of degree n, where n 1, then the equation f (x) 0 has at least one complex root. 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. The Fundamental Theorem of Algebra

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The Linear Factor Theorem If f (x) a n x n a n 1 x n 1 … a 1 x a 0 b, where n 1 and a n 0, then f (x) a n (x c 1 ) (x c 2 ) … (x c n ) where c 1, c 2,…, c n 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. The Linear Factor Theorem If f (x) a n x n a n 1 x n 1 … a 1 x a 0 b, where n 1 and a n 0, then f (x) a n (x c 1 ) (x c 2 ) … (x c n ) where c 1, c 2,…, c n 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. 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.

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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, and i as zeros and such that f (3) 150. Solution Because i is a zero and the polynomial has real coefficients, the conjugate must also be a zero. We can now use the Linear Factorization Theorem. a n (x 2)(x 2)(x i)(x i) Use the given zeros: c 1 2, c 2 2, c 3 i, and, from above, c 4 i. f (x) a n (x c 1 )(x c 2 )(x c 3 )(x c 4 ) This is the linear factorization for a fourth-degree polynomial. a n (x 2 4)(x 2 1) Multiply f (x) a n (x 4 3x 2 4) Complete the multiplication more

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

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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)

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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) = -16x 2 + 58x - 104

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