3.4 Zeros of Polynomial Functions• Obj: Find x-intercepts of a polynomial• Review of zeros

A ZERO is an X-INTERCEPT• Multiple Zeros

the zeros are x = 5 (mult of 2), -2 (mult of 3), and -4 (mult of 1)

A polynomial function of degree n has at most n zeros, where each zeros of multiplicity k is counted k times.

2 3( ) ( 5) ( 2) ( 4)f x x x x

The Factor TheoremA polynomial function P(x) has a factor (x – c) if and

only if P(c) = 0.

Determine whether the given binomial is a factor of P(x).

Use synthetic division. If the remainder is zero, the binomial is a factor. Otherwise, the binomial is not a factor.

3 2( ) 4 27 90, 6P x x x x x

Rational Zero TheoremIf

then the possible rational zeros are the factors of

Example: Use the Rational Zero Theorem to list possible rational zeros for the polynomial function

Possible Zeros =

1( ) ...n nf x qx x p

p

q

3 2( ) 2 9 2 9f x x x x

9

2

factors of

factors of

1,3,9 1 3 91,3,9, , ,

1, 2 2 2 2

PracticeUse the Rational Zero Theorem to list possible

rational zeros for the polynomial function

Find the zeros of the function.Use the list generated above to find the actual zeros.

Use the graphing calculator and synthetic division to find the zeros.

3 2( ) 2 25 12P x x x x

1,2,3,4,6,12 1 31,2,3,4,6,12, ,

1,2 2 2

p

q

Fundamental Theorem of Algebra

If P(x) is a polynomial function of degree n > 1, then p(x) has at least one complex zero.Remember, complex numbers are a + bi!

If P(x) is a polynomial function of degree n > 1, then P(x) has exactly n complex zeros.

Find the zeros. (next slide)

4 3 2( ) 4 8 16 16P x x x x x

ExampleFind the zeros of P(x) = x4 – 4x3 +8x2 – 16x + 16

Because the degree is 4, there are exactly 4 zeros.To find them, first list the possible rational zeros.

+1, 2, 4, 8, 16 so there are 10 possible rational zeros!

Use the graphing calculator to find the first zero.It looks like it touches at 2, so we use synthetic division to test.

Practice

Find all the zeros of the polynomial function.

List the possible rational zeros.Use the graphing calculator to

narrow the list.Use synthetic division to

reduce the polynomial and solve.

The three zeros are 5, 4 + 3i, and 4 – 3i.

3 2( ) 13 65 125P x x x x

Conjugate Pair Theorem

If a + bi is a complex zero of a polynomial function, then the conjugate a – bi is also a complex zero.

Use the given zero to find the remaining zeros of the function.

Use synthetic division and the given conjugate pair to find the other zeros.

3 2( ) 3 29 92 34;5 3P x x x x i

Find the ZerosBecause 5 + 3i is a zero, 5 – 3i is also a zero – Conjugate Pair

Theorem – so use synthetic division with both.

So the zeros are -1/3, 5 + 3i, and 5 – 3i.

Find a Polynomial Given ZerosFind a polynomial function of lowest degree

with integer coefficients that has the given zeros: 4, -3, 2.

(x – 4)(x + 3)(x – 2) 1. Write the zeros as linear factors.f(x) = x3 – 3x2 + x + 24 2. Multiply.

Zeros: 3, 2i, -2i Find the polynomial of lowest degree with integer coefficients that has thegiven zeros.

f(x) = x3 – 3x2 + 4x - 12

Assignment

•3.4 5 – 50 by 5, skip 35