7.5 and 7.6

Roots and Zeros of a Function

Zeros, Factors & RootsZeros, Factors & RootsSummary:Summary:

– cc is a zero of is a zero of f(x)f(x)

– x - cx - c is a factor of is a factor of f(x)f(x)

– cc is a root/solution of is a root/solution of f(x)f(x) = 0 = 0

– If If cc is real, is real, (c,0)(c,0) is an is an x-x-intercept.intercept.

1 2 11 2 1 0 Let ( ) ... .n n

n nf x a x a x a x a x a

Fundamental Theorem of AlgebraFundamental Theorem of Algebra

Every polynomial equation with degree Every polynomial equation with degree greater than zero has at least one greater than zero has at least one

root in the set of complex #’s.root in the set of complex #’s.

An An nnthth degree polynomial equation of the form degree polynomial equation of the form P(x) = 0P(x) = 0

1.1. has exactly has exactly nn roots in the set of complex #’s. roots in the set of complex #’s.

2.2. has exactly has exactly nn zeros. zeros.

Rational Zero TheoremRational Zero Theorem

can help us limit the # of possible zeros to testcan help us limit the # of possible zeros to test

– p is a factor of the constantp is a factor of the constant– q is a factor of the leading coefficientq is a factor of the leading coefficient

pIf is a rational number and a zero of ( ),

qf x

Ex 1Ex 1

List all the possible rational rootsList all the possible rational roots..f(x) = xf(x) = x44 + 7x + 7x33 – 15 – 15

g(x) = 6xg(x) = 6x33 + 6x + 6x22 – 15x - 2 – 15x - 2

EX 2EX 2Find all the zeros.Find all the zeros.

f(x) = xf(x) = x4 4 + x+ x33 - 19x - 19x22 + 11x + 30 + 11x + 30

Complex Conjugates TheoremComplex Conjugates Theorem

if if a + bia + bi is a zero, then is a zero, then a - bia - bi is a zero is a zero

Irrational Root TheoremIrrational Root Theorem

If is a root then is also a root.a b a b

EX 3EX 3Find all the zeros.Find all the zeros.

f(x) = xf(x) = x33– 4x– 4x22 + 6x – 4 + 6x – 4

EX 4EX 4Find all the zeros.Find all the zeros.

f(x) = 3xf(x) = 3x33 - 5x - 5x22 - 14x - 4 - 14x - 4

EX 5EX 5Find all the zeros.Find all the zeros.

f(x) = 6xf(x) = 6x44– 5x– 5x33 - 38x - 38x22 - 5x + 6 - 5x + 6

EX 6EX 6Find all the zeros.Find all the zeros.

f(x) = 6xf(x) = 6x44– 17x– 17x33 + 8x + 8x22 + 8x - 3 + 8x - 3

Ex 7Ex 7Write the polynomial function of least Write the polynomial function of least

degree w/integer coefficients whose zeros degree w/integer coefficients whose zeros include the following:include the following:

4 and 4 i

Ex 8Ex 8Write the polynomial function of least Write the polynomial function of least

degree w/integer coefficients whose zeros degree w/integer coefficients whose zeros include the following:include the following:2 and 1 i