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Finding Rational Zeros

Zeros = Solutions = Roots = x-intercepts

Find all zeros of x2 –10x + 24

You just factor and set each factor to zero. (You knew this already!)

(x – 12)(x + 2) = 0 x = 12, -2

You could also graph and see where it crosses the x-axis (x-intercepts) (You knew this already too!)

Refresh. What is a Rational Number?

A number that can be written as the ratio of two integers.

Examples: 31

197

15311

It can also be an ending or repeating decimal.

Examples: 6.21212121… 9.58310562 0.3333333…

Rational Zero Theorem

If f(x) = anxn + … + a1x + a0 (it’s a polynomial)

and the polynomial has integer coefficients, then

EVERY rational zero of f has the following form:

qp

factor of the constant term . factor of the leading coefficient

=

“p over q”

Find the rational zeros of f(x) = x3 + 2x2 – 11x – 12

p is all of the factors of the constant term.

1, 3,2, 4, 6, 12

q is all of the factors of the leading coefficient.

This one is easy because the leading coefficient is 1 !

The only factor is: 1

qp

= 1, 3,2, 4, 6, 12

Using qp

Still finding the rational zeros of f(x) = x3 + 2x2 – 11x – 12

1, 3,2, 4, 6, 12 Possible Zeros:

Do synthetic division until you find a zero.

x 3 + 2 x

2 + (-11) x + (–12)

1 3

1

-8

3

-20

-8

1 2 –11 - 121

k-value

1 •

Remainder is not z

ero so

1 is not a

zero to

this f

unction.

Test x = -1.

Keep trying Possible Zeros

x 3 + 2 x

2 + (-11) x + (–12)

1 1

-1

-12 0

12

1 2 –11 - 12-1

k-value

-1 •

Remainder IS ze

ro so

-1 IS a ze

ro to th

is functi

on!

Since -1 is a zero of f, then the result is a factor. (x2 + x – 12)

This is factorable into: (x + 4)(x – 3).

The zeros are: -1 (original zero), -4, 3

-1

The Nightmare Example

Find the rational zeros of f(x) = 10x4 - 3x3 - 29x2 + 5x + 12

qp

= 1, 3,2, 4, 6, 12

1 11 1 1 1

1, 3,2, 4, 6, 12

2 22 2 2 2

1, 3,2, 4, 6, 12

5 55 5 5 5

1, 3,2, 4, 6, 12

10 1010 10 10 10

The Nightmare Example (cont’d)

Finding the zeros of f(x) = 10x4 - 3x3 - 29x2 + 5x + 12

With so many possible zeros, it’s worth our time to get a ballpark answer by graphing the polynomial on the calculator.

10 -18

-15

-2

27

8

3 -12

0

10 -3 -29 5 12 -3/2 •

We found the 1st Zero!

What do you need to remember?

If f(x) = anxn + … + a1x + a0 (it’s a polynomial)

and the polynomial has integer coefficients, then

EVERY rational zero of f has the following form:

qp

factor of the constant term . factor of the leading coefficient

=

Rational Zero Theorem

Be able to list all possible rational zeros.Then let your calculator do the rest!