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1 Quadratic Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I

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A polynomial function of degree n is where the a’s are real numbers and the n’s are nonnegative integers and a n 0. Polynomial Function

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A polynomial function of degree 2 is called a quadratic function. It is of the form a, b, and c are real numbers and a 0. Quadratic Function

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For a quadratic function of the form gives the axis of symmetry. Axis of Symmetry

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A quadratic function of the form is in standard form. axis of symmetry: x = h vertex: (h, k) Standard Form

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Characteristics of Parabola a > 0 a < 0 vertex: minimum vertex: maximum

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7 Higher Degree Polynomial Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I

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The graph of a polynomial function… 1. Is continuous. 2. Has smooth, rounded turns. 3. For n even, both sides go same way. 4. For n odd, sides go opposite way. 5. For a > 0, right side goes up. 6. For a < 0, right side goes down. Characteristics

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. a n < 0 graphs of a polynomial function for n odd: Leading Coefficient Test: n odd a n > 0

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. a n < 0 graphs of a polynomial function for n even: a n > 0 Leading Coefficient Test: n even

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The following statements are equivalent for real number a and polynomial function f : 1. x = a is root or zero of f. 2. x = a is solution of f (x) = (x - a) is factor of f (x). 4. (a, 0) is x-intercept of graph of f (x). Roots, Zeros, Solutions

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1. If a polynomial function contains a factor (x - a) k, then x = a is a repeated root of multiplicity k. 2. If k is even, the graph touches (not crosses) the x-axis at x = a. 3. If k is odd, the graph crosses the x-axis at x = a. Repeated Roots (Zeros)

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If a < b are two real numbers and f (x)is a polynomial function with f (a) f (b), then f (x) takes on every real number value between f (a) and f (b) for a x b. Intermediate Value Theorem

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Let f (x) be a polynomial function and a < b be two real numbers. If f (a) and f (b) have opposite signs (one positive and one negative), then f (x) = 0 for a < x < b. NOTE to Intermediate Value

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15 Polynomial and Synthetic Division Dr. Claude S. Moore Danville Community College PRECALCULUS I

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If f (x) and d(x) are polynomials with d(x) 0 and the degree of d(x) is less than or equal to the degree of f(x), then q(x) and r (x) are unique polynomials such that f (x) = d(x) ·q(x) + r (x) where r (x) = 0 or has a degree less than d(x). Full Division Algorithm

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f (x) = d(x) ·q(x) + r (x) dividendquotient divisor remainder where r (x) = 0 or has a degree less than d(x). Short Division Algorithm

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ax 3 + bx 2 + cx + d divided by x - k kabcd ka ar coefficients of quotient remainder 1. Copy leading coefficient. 2. Multiply diagonally. 3. Add vertically. Synthetic Division

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If a polynomial f (x) is divided by x - k, the remainder is r = f (k). Remainder Theorem

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A polynomial f (x) has a factor (x - k) if and only if f (k) = 0. Factor Theorem

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21 Real Zeros of Polynomial Functions Dr. Claude S. Moore Danville Community College PRECALCULUS I

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a’s are real numbers, a n 0, and a 0 Number of positive real zeros of f equals number of variations in sign of f(x), or less than that number by an even integer. 2. Number of negative real zeros of f equals number of variations in sign of f(-x), or less than that number by an even integer. Descartes’s Rule of Signs

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a’s are real numbers, a n 0, and a 0 f(x) has two change-of-signs; thus, f(x) has two or zero positive real roots. 2. f(-x) = -4x 3 5x has one change-of- signs; thus, f(x) has one negative real root. Example 1: Descartes’s Rule of Signs

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Factor out x; f(x) = x(4x 2 5x + 6) = xg(x) 1. g(x) has two change-of-signs; thus, g(x) has two or zero positive real roots. 2. g(-x) = 4x 2 + 5x + 6 has zero change-of- signs; thus, g(x) has no negative real root. Example 2: Descartes’s Rule of Signs

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If a’s are integers, every rational zero of f has the form rational zero = p/q, in reduced form, and p and q are factors of a 0 and a n, respectively. Rational Zero Test

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f(x) = 4x 3 5x p { 1, 2, 3, 6} q { 1, 2, 4} p/q { 1, 2, 3, 6, 1/2, 1/4, 3/2, 3/4} represents all possible rational roots of f(x) = 4x 3 5x Example 3: Rational Zero Test

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f(x) is a polynomial with real coefficients and a n > 0 with f(x) (x - c), using synthetic division: 1. If c > 0 and each # in last row is either positive or zero, c is an upper bound. 2. If c < 0 and the #’s in the last row alternate positive and negative, c is an lower bound. Upper and Lower Bound

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2x 3 3x 2 12x + 8 divided by x c = -3 < 0 and #’s in last row alternate positive/negative. Thus, x = -3 is a lower bound to real roots. Example 4: Upper and Lower Bound

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