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!"#\$%# Lesson 7-5 and 7-6

Zeros, Factors, and Roots Let f(x) be a polynomial function. Suppose c is a zero of f(x). Then x-c is a factor of the polynomial f(x) and c is a root (or solution) of the equation f(x) = 0. Additionally, the ordered pair (c,0) is an x-intercept of the graph of f(x).

Fundamental Theorem of Algebra Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.

Ex. 1. Solve each equation. State the number and type of roots. a. x + 3 = 0 b. x2 - 8x +16 = 0 c. x3 + 2x = 0

Corollary of the Fundamental Theorem of Algebra A polynomial equation of the form P(x) = 0 of degree n with complex coefficients has exactly n roots in the set of complex numbers, provided that one counts double roots as two roots, triple roots as three roots, etc.

Note: Imaginary roots always occur in conjugate pairs.

Descartes' Rule of Signs If P(x) is a polynomial with real coefficients whose terms are arranged in descending powers of the variable, *the number of positive real zeros of y=P(x) is the same as the number of changes in sign of the coefficients of the terms, or is less than this by an even number, and *the number of negative real zeros of y=P(x) is the same as the number of changes in sign of the coefficients of the terms of P(-x), or is less than this be an even number.

Ex. 2. State the possible number of positive real zeros, negative real zeros, and imaginary zeros of p(x) = x5 - 6x4 - 3x3 + 7x2 - 8x + 1.

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# Ex. 3. Find all zeros of f(x) = x3 - 4x2 + 6x - 4.# # # # # # # # # # # Ex. 4. Write a polynomial function of least degree with integral coefficients that has the given zeros. 3, - 2i# # # # # # # # # # # # #

Rational Zero Theorem Let f(x) be a polynomial function with integral coefficients. If p/q is a rational number in simplest form and is a zero of y = f(x), then p is a factor of the constant term and q is a factor of the leading coefficient. Ex. 5. List all possible rational zeros of each function. a. f(x) = x4 - 3x3 + 5x2 - 2x + 6 b. f(x) = 2x3 + 3x2 - 17x + 12

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Ex. 6. For each of the polynomial functions: 1. List all possible rational zeros 2. Find all of the rational zeros. a. f(x) = x4 - 5x2 + 4 b. h(x) = 10x3 - 17x2 - 7x + 2# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # !"#\$%&'()%*!#+%,-%.*"*/%011+%0234%5!6%074\$%.8+%59+%*8% !"9\$%&'()%*8.+%,-%.*"**%011%%:;3-4%'<<%70--3=<)%>'430?'<%@)>0-%0?%'<<%7>0=<)2-A6%074\$%.B+%5C+%*5#

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/Al._2_Sections_75_and_76