6.6 Finding Rational Zeros pg. 359! What is the rational zero theorem? What information does it give you?

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6.6 Finding Rational Zeros pg. 359! What is the rational zero theorem? What information does it give you?

The rational zero theorem If f(x)=a n x + +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 n … =

Example 1: Find rational zeros of f(x)=x 3 +2x 2 -11x-12 1.List possible LC=1 CT=-12 X= ±1/1,± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1 2.Test: X= x= 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

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

Example 2: f(x)=10x 4 -3x 3 -29x 2 +5x+12 1.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 2.w/ so many –sketch graph on calculator and find reasonable solutions: x= -3/2, -3/5, 4/5, 3/2 Check: x= -3/ 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 __ ____

Repeat finding zeros for: g(x)=5x 3 -9x 2 -x+4 1. LC=5 CT=4 x:±1, ±2, ±4, ±1/5, ±2/5, ±4/5 *The graph of original shows 4/5 may be: x=4/ (2x+3)(x-4/5)(5x 2 -5x-5)= (2x+3)(x-4/5)(5)(x 2 -x-1)= mult.2 nd factor by 5 (2x+3)(5x-4)(x 2 -x-1)= -now use quad for last- *-3/2, 4/5, 1±, __ ____

What is the rational zero theorem? If f(x)=a n x + +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 What information does it give you? It gives you a pool of numbers to use to help you find a divisor.

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