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Finding Rational Zeros: Theorem and Examples

Learn how to find rational zeros of polynomial functions with integer coefficients using the rational zero theorem. This tutorial provides step-by-step examples and tests to determine the possible rational zeros, factor polynomials, and find solutions. Practice with different polynomial functions and understand the process of identifying rational zeros effectively. Helpful for students studying algebraic concepts.

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Finding Rational Zeros: Theorem and Examples

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  1. Finding Rational Zeros6.6pg. 359!

  2. The rational zero theorem … • If f(x)=anx + +a1x+a0 has integer coefficients, then every rational zero of f has the following form: pfactor of constant term a0 q factor of leading coefficient an n =

  3. Example 1: • Find rational zeros of f(x)=x3+2x2-11x-12 • List possible LC=1 CT=-12 X= ±1/1,± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1 • 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 • Since -1 is a zero: (x+1)(x2+x-12)=f(x) Factor: (x+1)(x-3)(x+4)=0 x=-1 x=3 x=-4

  4. Extra Example 1: • Find rational zeros of: f(x)=x3-4x2-11x+30 • LC=1 CT=30 x= ±1/1, ± 2/1, ±3/1, ±5/1, ±6/1, ±10/1, ±15/1, ±30/1 • 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)(x2-2x-15)=0 2 -4 -30 (x-2)(x+3)(x-5)=0 1 -2 -15 0 x=2 x=-3 x=5

  5. Example 2: • f(x)=10x4-3x3-29x2+5x+12 • List: LC=10 CT=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 • w/ so many –sketch 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)(10x3-18x2-2x+8)* (x+3/2)(2)(5x3-9x2-x+4) -factor out GCF (2x+3)(5x3-9x2-x+4) -multiply 1st factor by 2 ____ __

  6. Repeat finding zeros for: • g(x)=5x3-9x2-x+4 • LC=5 CT=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)(5x2-5x-5)= (2x+3)(x-4/5)(5)(x2-x-1)= mult.2nd factor by 5 (2x+3)(5x-4)(x2-x-1)= -now use quad for last- *-3/2, 4/5, 1± ,1- . 2 2 ____ __

  7. Assignment 362-363/18-57 mult of 3

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