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A 3 3.4 Zeros of Polynomial Functions

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A3 3.4 Zeros of Polynomial Functions

Homework: p. 387-388 1-31 eoo, 39-51 odd

Rational Zeros Theorem

Real zeros of polynomial functions are either rational

zeros or irrational zeros. Examples:

The function has rational zeros –3/2 and 3/2

The function has irrational zeros – 2 and 2

Rational Zeros Theorem

Suppose f is a polynomial function of degree n > 1 of the form

with every coefficient an integer and . If x = p /q is

a rational zero of f, where p and q have no common integer

factors other than 1, then

- p is an integer factor of the constant coefficient , and

- q is an integer factor of the leading coefficient .

RZT – Examples:

Find the rational zeros of

The leading and constant coefficients are both 1!!!

The only possible rational zeros are 1 and –1…check them out:

So f has no rational zeros!!!

(verify graphically?)

RZT – Examples:

Find the rational zeros of

Potential Rational Zeros:

Factors of –2

Factors of 3

Graph the function to narrow the search…

Good candidates: 1, – 2, possibly –1/3 or –2/3

Begin checking these zeros, using synthetic division…

RZT – Examples:

Find the rational zeros of

1

3

4

–5

–2

Because the remainder is zero,

x – 1 is a factor of f(x)!!!

3

7

2

3

7

2

0

Now, factor the remaining quadratic…

The rational zeros are 1, –1/3, and –2

RZT – Examples:

Find the polynomial function with leading coefficient 2 that has

degree 3, with –1, 3, and –5 as zeros.

First, write the polynomial in factored form:

Then expand into standard form:

RZT – Examples:

Using only algebraic methods, find the cubic function with the

given table of values. Check with a calculator graph.

x

–2

–1

1

5

(x + 2), (x – 1), and (x – 5)

must be factors…

f(x)

24

0

0

0

But we also have :

Properties of Roots of Polynomial Equations

A polynomial equation of degree n has n roots, counting repeated roots separately

If a+bi is a root to the polynomial equation with real coefficients ( ), then the imaginary number a-bi is also a root. Imaginary roots, if the occur, always occur in conjugate pairs

If f (x) is a polynomial function of degree n > 0, then

f (x) has precisely n linear factors and

where a is the leading coefficient of f (x), and

are the complex zeros of f (x).

the are not necessarily distinct numbers;

some may be repeated.

The following statements about a polynomial function f are

equivalent even if k is a nonreal complex number:

1. x = k is a solution (or root) of the equation f (x) = 0.

2. k is a zero of the function f.

3. x – k is a factor of f (x).

One “connection” is lost if k is a complex number…

k is not an x-intercept of the graph of f !!!

Fundamental Theorem of Algebra: if f(x) is a polynomial of degree n, then the equation f(x)=0 has at least one complex root.

Linear Factorization Theorem: given

then the linear factorization is:

Notice: the a’s are the same, and the linear factors are the zeros…

1. Find a 4th degree polynomial function with real coefficients that has zeros of -2, 2, and i such that f(3)=-150. Write the equation in factored form, and in general form.

2. Find an n-th degree polynomial function with real coefficients. Write the complete linear factorization and the polynomial in general form. n=3 (degree), x=6 and – 5+2i, f(2)=- 636