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6.6 – Finding Rational Zeros

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The Rational Zero Theorem If f(x) = a n x n + … + a 1 x + a 0 has integer coefficients, then every rational zero of f has the following form: p factor of constant term a 0 q factor of leading coefficient a n =

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Example 1: Find rational zeros of f(x) = x 3 + 2x 2 – 11x – 12 1.List possible factors Leading Coefficient = 1 Constant = -12 x = ±1/1,±2/1, ±3/1, ±4/1, ±6/1, ±12/1 2.Test: 1 2 -11 -12 1 2 -11 -12 x = 1 1 3 -8 x = -1 -1 -1 12 1 3 -8 -20 1 1 -12 0 3.Since -1 is a zero: (x + 1)(x 2 + x – 12) = f(x) Factor: (x + 1)(x – 3)(x + 4) = 0 x = -1 x = 3 x = -4

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Example 2: Find rational zeros of: f(x) = x 3 – 4x 2 – 11x + 30 1. Leading Coefficient = 1 Constant = 30 x = ±1/1, ±2/1, ±3/1, ±5/1, ±6/1, ±10/1, ±15/1, ±30/1 2.Test: 1 -4 -11 30 1 -4 -11 30 x = 1 1 -3 -14 x = -1 -1 5 6 1 -3 -14 16 1 -5 -6 36 x = 2 1 -4 -11 30 (x – 2)(x 2 – 2x – 15)=0 2 -4 -30 (x – 2)(x + 3)(x – 5)=0 1 -2 -15 0 x = 2 x = -3 x = 5

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f(x) = 10x 4 – 3x 3 – 29x 2 + 5x + 12 1.List: Leading Coefficient = 10 Constant = 12 x = ±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±12/1, ±3/2, ±1/5, ±2/5, ±3/5, ±6/5, ±12/5, ±1/10, ±3/10, ±12/10 2.With so many, graph on calculator and find reasonable solutions: x = -3/2, -3/5, 4/5, 3/2 Check: 10 -3 -29 5 12 x = -3/2 -15 27 3 -12 10 -18 -2 8 0 Yes, it works * (x + 3/2)(10x 3 – 18x 2 – 2x + 8) * (x + 3/2)(2)(5x 3 – 9x 2 – x + 4) - factor out GCF (2x + 3)(5x 3 – 9x 2 – x + 4) - multiply 1 st factor by 2 Example 3:

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Repeat Finding Zeros for: g(x) = 5x 3 – 9x 2 – x + 4 1. Leading Coefficient = 5 Constant = 4 x: ±1, ±2, ±4, ±1/5, ±2/5, ±4/5 *The graph of original shows 4/5 may be: 5 -9 -1 4 x = 4/5 4 -4 -4 5 -5 -5 0 (2x + 3)(x – 4/5)(5x 2 – 5x – 5) = (2x + 3)(x – 4/5)(5)(x 2 – x – 1) = multiply 2 nd factor by 5 (2x + 3)(5x – 4)(x 2 – x – 1) = -now use quadratic formula for last factor- -3/2, 4/5, 1±, 2

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Homework Assignment: pgs. 362-363 #15, 16, 23, 33, 34

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6.6 – Finding Rational Zeros (Day 2)

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Homework Check/Questions

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Refresher 1: Find the rational zeros of f(x) = x 3 + 4x 2 + 1x – 6 Factored form: (x + 3)(x – 1)(x + 2) Solutions (roots): x = -3 x = 1 x = -2

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Refresher 2: Find the rational zeros of f(x) = 3x 4 + 11x 3 + 11x 2 + x – 2 Factored form: (x + 2)(x + 1)(3x – 1)(x + 1) Solutions: x = 2 x = -1 x = 1/3

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Battleship Practice 1) Each partner will deploy their ships into battle. 2) Each partner will take turns guessing where the other person’s ships are at on his or her board. 3) When a hit is made, both students will solve the appropriate problem for the ship that was hit. 4) To sink any ship, you have to list all the possible roots, the correct factorization, and the actual roots for the given polynomial. 5) The person who made the hit first has to correctly complete Step #4 for in order to successfully sink the ship. 6) If the person who made the hit is wrong, the other person can steal the hit if their answer is correct. This will result in their partner’s ship being sunk. 7) If both partners are wrong or a dispute occurs about which partner got the correct answer first, an additional problem will be provided to solve.

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Homework Assignment: pgs. 362-363 #35-38

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EXAMPLE 3 Find zeros when the leading coefficient is not 1

EXAMPLE 3 Find zeros when the leading coefficient is not 1

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