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# Factoring Polynomials Help

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By McGraw-Hill Professional
Updated on Oct 4, 2011

## Introduction to Factoring Polynomials

### The Rational Zero Theorem

The Rational Zero Theorem says that if a polynomial function f(x) , with integer coefficients, has a rational number p/q as a zero, then p is a divisor of the constant term and q is a divisor of the leading coefficient. Not all polynomials have rational zeros, but most of those in precalculus courses do.

The Rational Zero Theorem is used to create a list of candidates for zeros. These candidates are rational numbers whose numerators divide the polynomial’s constant term and whose denominators divide its leading coefficient. Once we have this list, we will try each number in the list to see which, if any, are zeros. Once we have found a zero, we can begin to factor the polynomial.

#### Examples

List the possible rational zeros.

• f(x) = 4 x 3 + 6 x 2 − 2 x + 9

The numerators in our list will be the divisors of 9: 1, 3, and 9 as well as their negatives, − 1, −3, and −9. The denominators will be the divisors of 4: 1, 2, and 4. The list of possible rational zeros is—

This list could be written with a little less effort as , .

We only need to list the numerators with negative numbers and not the denominators. The reason is that no new numbers are added to the list, only duplicates of numbers already there. For example, and are the same number.

• g(x) = 6 x 4 − 5 x 3 + 2 x − 8

The possible numerators are the divisors of 8: ±1, ±2, ±4, and ±8. The possible denominators are the divisors of 6: 1, 2, 3, and 6. The list of possible rational zeros is—

There are several duplicates on this list. There will be duplicates when the constant term and leading coefficient have common factors. The duplicates don’t really hurt anything, but they could waste time when checking the list for zeros.

## Strategy for Factoring Polynomials

Now that we have a starting place, we can factor many polynomials. Here is the strategy. First we will see if the polynomial can be factored directly. If not, we need to list the possible rational zeros. Then we will try the numbers in this list, one at a time, until we find a zero. Once we have found a zero, we will use polynomial division (long division or synthetic division) to find the quotient. Next, we will factor the quotient. If the quotient is a quadratic factor, we will either factor it directly or use the quadratic formula to find its zeros. If the quotient is a polynomial of degree 3 or higher, we will need to start over to factor the quotient. Eventually, the quotient will be a quadratic factor.

#### Examples

Completely factor each polynomial.

• f(x) = 3 x 4 − 2 x 3 − 7 x 2 − 2 x

First we will factor x from each term: f(x) = x (3 x 3 − 2 x 2 − 7x − 2). The possible rational zeros for 3 x 3 − 2 x 2 − 7 x − 2 are ±1, ±2, .

3(1) 3 − 2(1) 2 − 7(1) − 2 ≠ 0

3(−1) 3 − 2(−1) 2 − 7(−1) −2 = 0

We will use synthetic division to find the quotient for (3 x 3 − 2 x 2 − 7 x − 2) ÷ ( x + 1).

The quotient is 3 x 2 − 5 x − 2 which factors into (3 x + 1)( x − 2).

f(x) = 3 x 4 − 2 x 3 − 7 x 2 − 2 x

= x (3 x 3 − 2 x 2 − 7 x − 2)

= x ( x + 1)(3 x 2 −5 x − 2)

= x ( x + 1)(3 x + 1)( x − 2)

• h(x) = 3 x 3 + 4 x 2 − 18 x + 5

The possible rational zeros are ±1, ±5, , and .

h (1) = 3(1 3 ) + 4(1 2 ) − 18(1) + 5 ≠ 0

h (−1) = 3(−1) 3 + 4(−1) 2 − 18(−1) + 5 ≠ 0

h (5) = 3(5 3) + 4(5 2) − 18(5) + 5 ≠ 0

Continuing in this way, we see that h (−5) ≠ 0, and .

We will find the zeros of x 2 + 3 x − 1 using the quadratic formula.

For a polynomial such as f(x) = 5 x 3 + 20 x 2 − 9 x − 36, the list of possible rational zeros is quite long—36! There are ways of getting around having to test every one of them. The fastest way is to use a graphing calculator to sketch the graph of y = 5 x 3 + 20 x 2 − 9 x − 36. There appears to be an x -intercept at x = −4 (remember that x -intercepts are zeros.)

f(x) = ( x + 4)(5 x 2 − 9) We will solve 5 x 2 − 9 = 0 to find the other zeros.

Practice problems for this concept can be found at: Polynomial Functions Practice Test.

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