finding rational zeros 6 6 pg 359 n.
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Finding Rational Zeros 6.6 pg. 359!. 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:.

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the rational zero theorem
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

=

example 1
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

extra example 1
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

example 2
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

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repeat finding zeros for
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

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