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Similarly, if x k is a factor of f ( x ) , then the remainder of the Division Algorithm f ( x ) = ( x k ) q ( x ) + r is 0. This tells us that k is a zero.

This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree n in the complex number system will have n zeros. We can use the Factor Theorem to completely factor a polynomial into the product of n factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.

The factor theorem

According to the Factor Theorem    , k is a zero of f ( x ) if and only if ( x k ) is a factor of f ( x ) .

Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.

  1. Use synthetic division to divide the polynomial by ( x k ) .
  2. Confirm that the remainder is 0.
  3. Write the polynomial as the product of ( x k ) and the quadratic quotient.
  4. If possible, factor the quadratic.
  5. Write the polynomial as the product of factors.

Using the factor theorem to solve a polynomial equation

Show that ( x + 2 ) is a factor of x 3 6 x 2 x + 30. Find the remaining factors. Use the factors to determine the zeros of the polynomial .

We can use synthetic division to show that ( x + 2 ) is a factor of the polynomial.

2 1 6 1 30 2 16 30       1 8 15      0

The remainder is zero, so ( x + 2 ) is a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:

( x + 2 ) ( x 2 8 x + 15 )

We can factor the quadratic factor to write the polynomial as

( x + 2 ) ( x 3 ) ( x 5 )

By the Factor Theorem, the zeros of x 3 6 x 2 x + 30 are –2, 3, and 5.

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Use the Factor Theorem to find the zeros of f ( x ) = x 3 + 4 x 2 4 x 16 given that ( x 2 ) is a factor of the polynomial.

The zeros are 2, –2, and –4.

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Using the rational zero theorem to find rational zeros

Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. But first we need a pool of rational numbers to test. The Rational Zero Theorem    helps us to narrow down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient    of the polynomial

Consider a quadratic function with two zeros, x = 2 5 and x = 3 4 . By the Factor Theorem, these zeros have factors associated with them. Let us set each factor equal to 0, and then construct the original quadratic function absent its stretching factor.

This image shows x minus two fifths equals 0 or x minus three fourths equals 0. Beside this math is the sentence, 'Set each factor equal to 0.' Next it shows that five x minus 2 equals 0 or 4 x minus 3 equals 0. Beside this math is the sentence, 'Multiply both sides of the equation to eliminate fractions.' Next it shows that f of x is equal to (5 x minus 2) times (4 x minus 3). Beside this math is the sentence, 'Create the quadratic function, multiplying the factors.' Next it shows f of x equals 20 x squared minus 23 x plus 6. Beside this math is the sentence, 'Expand the polynomial.' The last equation shows f of x equals (5 times 4) times x squared minus 23 x plus (2 times 3).

Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5 and 4.

We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient. This is the essence of the Rational Zero Theorem; it is a means to give us a pool of possible rational zeros.

The rational zero theorem

The Rational Zero Theorem    states that, if the polynomial f ( x ) = a n x n + a n 1 x n 1 + ... + a 1 x + a 0 has integer coefficients, then every rational zero of f ( x ) has the form p q where p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n .

When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.

Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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