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4.3 – Location Zeros of Polynomials

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At times, finding zeros for certain polynomials may be difficult There are a few rules/properties we can use to help us at least determine some rough information pertaining to the zeros

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Rational Zero Theorem For a polynomial f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0, then any rational zero must be of the form (p/q), where p is a factor of the constant term (a 0 ) and q is a factor of the leading coefficient a n – All zeros are ratios of the constant and leading coefficient

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Example. For the function f(x) = 2x 3 + 5x 2 – 4x - 3, list the potential rational zeros.

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Descartes’ Rule of Signs = for a polynomial, a variation in sign is a change in the sign of one coefficient to the next – 1) The number of positive real zeros is either the number of variation in sign or is less than this number by a positive integer (MAXIMUM, +) – 2) The number of negative real zeros is the number of variations in the sign of f(-x) or less than this by a positive integer (MAXIMUM, -)

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Example. For the function f(x) = 2x 3 + 5x 2 – 4x - 3, determine the number of possible positive and negative zeros. Example. For the function f(x) = x 3 – 6x 2 + 13x – 20, determine the number of possible positive and negative zeros.

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Intermediate Value Theorem = For a polynomial f(x), if f(a) and f(b) are different in signs, and a < b, then there lies at least one zero between a and b – Most useful!

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Example. Show that f(x) = x 3 + 3x – 7 has a zero between 1 and 2.

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Multiplicity of Zeros/Roots Suppose that c is a zero for the polynomial f(x) For (x – c) k, or a root of multiplicitity k, the following rules are known: – The graph of f will touch the axis at (c,0) – Cross through the x-axis if k is odd – Stay on the same side of the x-axis if k is even As k gets larger than 2, the graph will flatten out

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Example. Sketch the graph of: f(x) = (x + 2)(x + 1) 2 (x – 3) 3

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Conjugate Pairs Recall, if you have the complex number a + bi, then the conjugate is a – bi If a polynomial f(x) has the imaginary zero a + bi, then the polynomial also has the conjugate zero, a – bi – If x – (a + bi) is a factor, so is x – (a – bi)

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Example. Construct a fourth degree polynomial function with zeros of 2, -5, and 1 + i, such that f(1) = 12.

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Conclusion: Still best to use, when applicable, any graphing utility to find the zeros of functions. However, when you’re stuck, these properties give us a wide range of potential values to choose from, and methods to narrow down the list of “candidate zeros.”

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Assignment Pg. 337 For the problems, do the following: 1) List the potential zeros 2) Show # of + or – zeros 3) Use your graphing calculator to identify the zeros #58, 60, 64

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The Rational Zero Theorem

The Rational Zero Theorem

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