5.6 The Fundamental Theorem of Algebra. If P(x) is a polynomial of degree n where n > 1, then P(x) = 0 has exactly n roots, including multiple and complex.

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5.6 The Fundamental Theorem of Algebra

If P(x) is a polynomial of degree n where n > 1, then P(x) = 0 has exactly n roots, including multiple and complex roots

Fundamental Theorem of Algebra Degree of polynomial= # of solutions BUT…some are repeated or imaginary

Repeated Zeros f(x) = (x + 4)(x – 5) 2

Find all the zeros of f(x)=x 3 + 3x x +48

Example What are all the roots of x 5 – x 4 – 3x 3 + 3x 2 – 4x + 4 = 0

Example What are the zeros of f(x) = x 4 + x 3 – 7x 2 – 9x -18?

Complex Zeros The complex zeros of a polynomial function with real coefficients always occur in complex conjugate pairs. That is, if a + bi is a zero, then a – bi must also be a zero.

Example Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and 2, 3, and -1 as zeros.

Example Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and 1, 5 and 2i as zeros.

YOU TRY Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and 8 and i as zeros.

Example Approximate the real zeros of f(x)=x 4 – 2x 3 – x 2 – 2x -2 Use calculator Where are the rest  They are imaginary

Finding approximate zeros 1. f(x) = x 3 – 3x 2 – 2x + 6

Exit Ticket 1. What are all the roots of the equation x 4 + 2x 3 = 13x 2 – 10x 2. What are all the zeros of the functions g(x) = 2x 4 – 3x 3 – x – 6? 3. Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and 5 and -3i as zeros. 4. Find the approximate zeros: f(x) = x 3 – 4x 2 – 5x + 14