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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.

The Rational Zero Test
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!

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