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Published byDonald Lambert Burke Modified over 9 years ago
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5.6 The Fundamental Theorem of Algebra
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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
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Fundamental Theorem of Algebra Degree of polynomial= # of solutions BUT…some are repeated or imaginary
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Repeated Zeros f(x) = (x + 4)(x – 5) 2
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Find all the zeros of f(x)=x 3 + 3x 2 + 16x +48
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Example What are all the roots of x 5 – x 4 – 3x 3 + 3x 2 – 4x + 4 = 0
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Example What are the zeros of f(x) = x 4 + x 3 – 7x 2 – 9x -18?
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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.
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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.
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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.
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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.
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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
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Finding approximate zeros 1. f(x) = x 3 – 3x 2 – 2x + 6
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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
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