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5.8 Rational Zero Theorem

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T HE R ATIONAL Z ERO T HEOREM : If f(x) = a n x n + a n-1 x n-1 + a n-2 x n-2 + … + a 1 x 1 + a 0 has integer coefficients, then every rational zero of f(x) has the following form:

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F IND ALL RATIONAL ROOTS : 1. List all possible rational roots: 2. Use calculator to decide which roots to test. (Correct if remainder = 0) 3. Use synthetic division to find the unknown factor 4. Repeat step 2 with this new factor until you have a quadratic. 5. Factor the quadratic or use the quadratic formula. 6. Set unsolved factors to zero and solve

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1. x 3 – 4x 2 – 11x + 30 List all possible rational roots: Use calculator to decide which to test: Use synthetic division to get the polynomial down to a quadratic. Factor/Quad Formula and solve X = -3, 2, 5 E XAMPLES F IND ALL ZEROS OF THE POLYNOMIAL

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2. f(x) = x 3 – x 2 – 6x + 8

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E XAMPLES F IND ALL ZEROS OF THE POLYNOMIAL 3. f(x) = 15x 4 – 68x 3 – 7x 2 + 24x – 4

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E XAMPLES F IND ALL ZEROS OF THE POLYNOMIAL

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4. f(x) = x 3 – 7x 2 + 10x + 6

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E XAMPLE 5 The volume of a rectangular solid is 1120 cubic feet. The width is 2 feet less than the height, and the length is 4 feet more than the height. Find the dimensions of the solid.

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E XAMPLE 6 Some ice sculptures are made by filling a mold with water and then freezing it. You are making such an ice sculpture for a school dance. It is to be shaped like a pyramid with a height that is 1 foot greater than the length of each side of its square base. The volume of the ice sculpture is 4 cubic feet. What are the dimensions of the mold?

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E XAMPLE 7 You are building a solid concrete wheelchair ramp. The width of the ramp is three times the height, and the length is 5 feet more that 10 times the height. If 150 cubic feet of concrete is used, what are the dimensions of the ramp?

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