# Real Zeros of Polynomial Functions

In the previous sections , we covered a method for
factoring a polynomial

by testing potential zeros with synthetic division to see if the remainder is

zero. One problem: There is an infinite number of "potential" zeros to

test.

The Rational Zero Test narrows down the list of potential

rational zeros to check.

If f(x) is the polynomial

f(x) = ax^n + b x^(n-1) + . . . + cx^2 + dx + k ,

Then ALL rational zeros will be of the form p/q where

p is a factor of the constant term k , and

q is a factor of the leading coefficient a .

Example: Find the rational zeros of f(x) = x^4- x^3 -8x +
8

.

According to the Rational Zero Test, all rational zeros will have the form

x = p/q where p is a factor of 8 and q is a factor of 1 .

**p is one of the following: {1,-1, 2, -2 , 4, -4, 8, -8}
q is one of the following: {1, -1}**

The possibilities of p /q include:

1/1 = 1, 1/-1 = -1, -1/1 = -1, -1/-1 = 1, 2/-1 = -2, -2/-2 = 1,

4/1 = 4, 4/-1 = -4, -4/1 = -4, -4/-1 = 4, 8/1=8, 8/-1 = -8, -8/1 = -8, -8/-1 = 8

As you have probably noticed, many possibilities are
repeated.

The list of possibilities is summarized as {+/-1, +/-2, +/-4, +/-8}.

These 8 are POSSIBLE rational zeros. To see which, if any,
are actual zeros,

you must check with synthetic division .

Synthetic division using these 8 possible zeros reveals
that 2 of them are zeros

and the other 6 are not. x = 1 and x = 2 are zeros. The synthetic division for

these 2 values is given below

**Notes on The Rational Zero Test**

The Rational Zero Test only finds potential RATIONAL
zeros. A rational zero

is a zero that may be written as a positive or negative whole number, a
fraction,

a terminating decimal ( like 0.25), or a repeating decimal that represents a

fraction (like 0.3333. . . =1/3). This test DOESN'T indicate possible

IRRATIONAL zeros. An irrational zero is a zero that is a complex number

(like 3 + 2i) or a decimal that does not terminate or repeat with a pattern.

x= sqrt(2) = 1.414213566237. . .is irrational. So is (1 +sqrt(3))/2.

For example, f(x) = x^2+ x + 1 has possible rational zeros
of 1 and -1, but

neither are zeros. In this case, both zeros are irrational. In fact, they are

complex numbers.

**Completely Factoring a Polynomial**

We may use the rational zero test and the idea that the
factorization

is written in terms of zeros to completely factor f(x) = x^4-x^3- 8x +8.

Previously in this section, we found the possible rational
zeros to be

x = +/-1, +/-2, +/-4, and +/-8. Testing these resulted in only x = 1 and x = 2

as actual zeros.

This means that f(x) = (x - 1)(x - 2)( unknown quadratic
factor ). Thus, we

know that x^4 - x^3 - 8x + 8 = (x - 1)(x - 2)( unknown quadratic factor).

Since (x-1)(x-2) = x^2 - 3x + 2, we may also say that

x^4 - x^3 -8x + 8 =(x^2 -3x +2)(unknown quadratic factor).

We may long divide f(x) by x^2-3x +2 in order to find that
quadratic factor.

This division is shown on the next page.

*Note that we can't use synthetic division when dividing by a quadratic.
*

With this result, we may write

f(x) as

f(x) = (x-1)(x-2)(x^2 +2x+4).

To completely factor f(x), we must find the zeros of the quadratic x^2 +2x +4

by using the quadratic formula.

Since the 4 zeros of x^4 -x^3- 8x + 8 are

the factorization is

We may remove the extra parentheses and write this as

**Can I do This Problem Without Long Dividing?**

The answer is YES. Another, different approach to
factoring

f(x) = x^4 - x^3 - 8x + 8, uses synthetic division instead of long

polynomial division . The difference in this method is that we reduce

the polynomial as we check the possible zero with synthetic division.

See the next page. . . .

**Another Approach to Factoring x^4 -x^3 -8x+8**

We start checking the possible rational zeros +/-1,+/-2,+/-4, +/-8.

We start with x = 1. Using synthetic division, we obtain

The bottom row indicates the coefficients of

the result after dividing.

The result after dividing is 1x^3 +0x^2+0x - 8 or x^3 - 8.

At this point we know that f(x) = (x-1)(x^3 - 8).

Now, check for zeros in x^3 - 8. Checking x=2 with
synthetic division

results in

This means that x^3 - 8 = (x - 2) (x^2 + 2x + 4) Continued. . .

We may finish the factorization by finding the zeros of
the quadratic

x^2+2x + 4 with the quadratic formula, as was done using the first method .

In this method, we "break down" f(x) into a linear factor
multiplied by another

polynomial of lesser degree. We then break down that polynomial of lesser

degree by finding its zeros and factoring it. When we obtain a quadratic, we

may break it down by factoring it directly or using the quadratic formula to

find its zeros.

**Descartes Rule of Signs & Bound Rules for Real Zeros
**You are not responsible for this. These rules are important in a Senior-level

college math course for Math majors but may be excluded in this course.

Do the assigned homework for 3.4. Read the text!

Prev | Next |