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

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

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

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.

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

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   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

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)

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 x - 104