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Transcript
Pre-Calculus Section 3.5 Objective: SWBAT use the Fundamental Theorem of Algebra to solve equations with real and complex solutions. Homework: Page 298 #13, 19, 21 and 31 – 57 odd Daily Warm – Up Substitute the function f ( x) x3 2 x 2 3x 1 into the difference quotient 1. Fundamental Theorem of Algebra a. A polynomial function has at least one zero in the set of complex numbers. b. An nth -degree polynomial function has exactly n zeros in the set of complex numbers, counting multiple zeros. c. If a polynomial has only real coefficients, then any nonreal complex zeros appear in conjugate pairs. 2. Given P( x) x3 3x 2 x 3 find: a. All the zeros of P. b. The complete factorization of P. 3. Zeros and their multiplicity. Every polynomial of degree n 1has exactly n zeros, provided that a zero of multiplicity k is counted k times. 4. Find the complete factorization and all five zeros of the polynomial P( x) 3x5 24 x3 48x 5. Find a polynomial P ( x ) of degree 4, with zeros i, i, 2, and 2 and with P(3) 25 . 6. Find a polynomial Q ( x) of degree 4, with zeros -2 and 0, where -2 is a zero of multiplicity 3. 7. Find the zeros of P( x) 3x 4 2 x3 x 2 12 x 4 8. Complex zeros occur in conjugate pairs. Of a polynomial P has real coefficients, and if the complex number z is a zero of P, then its complex conjugate z is also a zero of P. 9. Find a polynomial P(x) of degree 3 that has integer coefficients and zeros 1 2 and (3 i ) . 10. Without factoring, determine how many positive real zeros, negative real zeros, and imaginary zeros the following polynomial could have: P( x) x 4 6 x3 12 x 2 14 x 24