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1. Describe the end behavior of the graph y = 2x 5 – 3x 2 + 5. 2. Sketch a graph of 3 rd degree with a zero at -5 (multiplicity 2) and a zero at 0 (multiplicity 1). 1

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Pre-Cal Real Zeros of Polynomial Functions.

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To solve, you factor the problem, each factor must be factored down to at most a quadratic. To solve, put each factor equal to zero. 3

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1. 2x 5 + 24x = 14x 3 2x 5 – 14x 3 + 24x = 0 2x(x 4 – 7x 2 + 12) = 0 2x(x 2 – 3)(x 2 – 4) = 0 2x=0 x 2 –3=0 x 2 –4=0 x=0 x 2 =3 x 2 = 4 x=±√3 x = ±2 2. 2y 5 – 18y = 0 2y(y 4 – 9) = 0 2y(y 2 – 3)(y 2 + 3) = 0 2y=0 y 2 –3=0 y 2 +3=0 y=0 y 2 = 3 y 2 = –3 y=±√3 y=±i√3 4

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Long Division Synthetic Division 5

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Divide 3x 3 – 5x 2 + 10x – 3 by 3x + 1 3x + 1 ) 3x 3 – 5x 2 + 10x – 3 6 x2x2 3x 3 + x 2 -6x 2 + 10x - 2x -6x 2 - 2x 12x - 3 + 4 12x + 4 -7 -7. 3x + 1 +

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Divide 2x 3 – 7x 2 – 17x – 3 by 2x + 3 2x + 3 ) 2x 3 – 7x 2 – 17x – 3 7 x2x2 2x 3 + 3x 2 -10x 2 - 17x - 5x -10x 2 - 15x - 2x - 3 - 1 -2x - 3 0

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Divide 6x 2 + x – 7 by 2x + 3 2x + 3 ) 6x 2 + x – 7 8 3x 6x 2 + 9x -8x- 7 - 4 - 8x - 12 5 5. 2x + 3 +

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Divide 4x 3 + 5x 2 - 3x + 10 by x 2 + 3 x 2 + 0x + 3 ) 4x 3 + 5x 2 – 3x + 10 9 4x 4x 3 + 0x 2 + 12x 5x 2 – 15x + 10 + 5 5X 2 + 0x + 15 -15x – 5 x 2 + 3

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Divide -3x 3 + 4x – 1 by x - 1 x - 1 ) -3x 3 + 0x 2 + 4x – 1 10 -3x 2 -3x 3 + 3x 2 -3x 2 + 4x - 3x -3x 2 + 3x 1x - 1 + 1 1x - 1 0

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Divide a polynomial by a binomial The polynomial must be in standard form Be sure to add zeros when needed for missing placeholders The divisor must be a linear binomial with a leading coefficient of 1 and in the form x + k 11

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12 f(x) = x 3 + 2x 2 – 3x – 6 ; x + 2 is a factor x = -2 -2 1 2 - 3 -6 times 10 0 -3 6 0 Now solve x 2 – 3 = 0 to find the remaining zeros. -2 ++

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13 f(x) = 6x 3 – 19x 2 + 16x – 4 ; x – 2 is a factor. X = 2 2 6 -19 16 -4 times 12 6-7 -14 2 4 0 Now solve 6x 2 – 7x + 2 = 0 +++

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16 F(x) = 2x(x - 3)(x + 5) = 2x 3 + 4x 2 - 30x

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V = x 3 + 9x 2 – x – 105 Given sides: x – 3 and x + 5 17 3 3rd side = x + 7

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Divide 6x 3 + 11x 2 – 12x – 13 by 3x + 1 using long division. Divide 10x 3 – 10x 2 – 2x + 4 by x + 3 using synthetic division. Factor 4x 3 – 3x 2 – 25x – 6 using synthetic division given that one zero is -2. Solve 6x 3 + 23x 2 – 33x + 10 given that one zero is -5. 18

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