2.6 Find Rational Zerospg. 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-121. List possible LC=1 CT=-12 X= ±1/1,± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/12. Test: 1 2 -11 -12 1 2 -11 -12X=1 1 3 -8 x=-1 -1 -1 12 1 3 -8 -20 1 1 -12 03. 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, ± 12Factors of the leading coefficient: ± 1

Simplified list of possible zeros: ± 1, ± 2, ± 3, ± 4, ± 6, ± 12

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+301. LC=1 CT=30x= ±1/1, ± 2/1, ±3/1, ±5/1, ±6/1, ±10/1, ±15/1, ±30/12.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

1 1 – 8 11 20Test x =1:

1 – 7 41 – 7 4 24

Test x = –1:–1 1 –8 11 20

1 – 9 20 0 –1 9 20

1 is not a zero.

–1 is a zero

Test these zeros using synthetic division.

Because –1 is a zero of f, you can write f (x) = (x + 1)(x2 – 9x + 20).

STEP 3

f (x) = (x + 1) (x2 – 9x + 20)

Factor the trinomial in f (x) and use the factor theorem.

The zeros of f are –1, 4, and 5.ANSWER

= (x + 1)(x – 4)(x – 5)

Find Zeros -leading coefficient is not 1 • f(x)=10x4-3x3-29x2+5x+121. 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, ± 12/102. 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 3If 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.

=