7.6 rational zero theorem algebra ii w/ trig. rational zero theorem: if a polynomial has integer...

7.6 Rational Zero Theorem Algebra II w/ trig

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7.6 Rational Zero TheoremAlgebra II w/ trig

•RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be a factor of the constant term divided by a factor of the leading coefficient.▫For

▫Constant term: number hanging off the end

▫Leading coefficient: an

•Remember roots and zeros are the solutions to the equation f(x)=0

11 1 0( ) n n

n nf x a x a x a x a

Page 3: 7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be

I. List all of the possible rational zeros of each function.

A. 4 2( ) 3 2 6 10

1, 2, 5, 10

1, 3

1 2 5 101, 2, 5, 10, , , ,

3 3 3 3

f x x x x

Page 4: 7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be

4 3( ) 3 2 5

1, 5

1, 3

1 51, 5, ,

3 3

h x x x

3 2( ) 10 14 36

1, 2, 3, 4, 6, 9, 12, 18

g x x x x

Page 5: 7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be

II. Find all zeros.

A. 3 2( ) 5 12 29 12f x x x x

Page 6: 7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be

B. 3 2( ) 4 2 20g x x x x

Page 7: 7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be

C. f(x) = 8x4 + 2x3 + 5x2 + 2x - 3

Page 8: 7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be

D. g(x) = x4 + 2x3 – 11x2 - 60

Page 9: 7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be

E. f(x)= x5 – 6x3 + 8x

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