2.6 Find Rational Zeros pg. 128 What is the rational zero theorem? What information does it give you?

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

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

List the possible rational zeros of f using the rational zero theorem. a. f (x) = x3 + 2x2 – 11x + 12 Factors of the constant term: ± 1, ± 2, ± 3, ± 4, ± 6, ± 12 Factors of the leading coefficient: ± 1   Simplified list of possible zeros: ± 1, ± 2, ± 3, ± 4, ± 6, ± 12

Factors of the constant term: + 1, + 2, + 5, + 10 f (x) = 4x4 – x3 – 3x2 + 9x – 10 Factors of the constant term: + 1, + 2, + 5, + 10 Factors of the leading coefficient: + 1, + 2, + 4    

Extra Example: Find rational zeros of: f(x)=x3-4x2-11x+30 LC=1 CT=30 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

Find all real zeros of f (x) = x3 – 8x2 +11x + 20. SOLUTION STEP 1  

STEP 2 Test these zeros using synthetic division. 1 1 – 8 11 20 Test x =1: 1 – 7 4 1 – 7 4 24 1 is not a zero. Test x = –1: –1 1 –8 11 20 1 – 9 20 0 –1 9 20 –1 is a zero

Because –1 is a zero of f, you can write f (x) = (x + 1)(x2 – 9x + 20). STEP 3 Factor the trinomial in f (x) and use the factor theorem. f (x) = (x + 1) (x2 – 9x + 20) = (x + 1)(x – 4)(x – 5) The zeros of f are –1, 4, and 5. ANSWER

Assignment 2.6 p. 132, 1, 4-22 even, 36-37

Find Zeros -leading coefficient is not 1 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

Repeat finding zeros for:    

If the highest degree is more than 3 If the highest degree is more than 3 (like 4) you will need to do synthetic division again, this time on the “new” equation you just found. Your goal is to divide your equation down to a 2nd degree equation so you can factor or use the quadratic formula. Each time you do synthetic division, your equation goes down 1 degree.

What is the rational zero theorem? If f(x)=anx + +a1x+a0 has integer coefficients, then every rational zero of f has the following form: p factor of constant term a0 q factor of leading coefficient an What information does it give you? It gives you a pool of numbers to use to help you find a divisor. =

Assignment 2.6 p. 132, 24-34 even, 41-43 all