Use the Factor Theorem to solve a polynomial equation

The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm tells us

f(x)=(xk)q(x)+rf\left(x\right)=\left(x-k\right)q\left(x\right)+r
.

If k is a zero, then the remainder r is

f(k)=0f\left(k\right)=0
 and
f(x)=(xk)q(x)+0f\left(x\right)=\left(x-k\right)q\left(x\right)+0
 or
f(x)=(xk)q(x)f\left(x\right)=\left(x-k\right)q\left(x\right)
.

Notice, written in this form, x – k is a factor of

f(x)f\left(x\right)
. We can conclude if is a zero of
f(x)f\left(x\right)
, then
xkx-k
is a factor of
f(x)f\left(x\right)
.

Similarly, if

xkx-k
 is a factor of
f(x)f\left(x\right)
, then the remainder of the Division Algorithm
f(x)=(xk)q(x)+rf\left(x\right)=\left(x-k\right)q\left(x\right)+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.

A General Note: The Factor Theorem

According to the Factor Theorem, k is a zero of

f(x)f\left(x\right)
 if and only if
(xk)\left(x-k\right)
 is a factor of
f(x)f\left(x\right)
.

How To: 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
    (xk)\left(x-k\right)
    .
  2. Confirm that the remainder is 0.
  3. Write the polynomial as the product of
    (xk)\left(x-k\right)
    and the quadratic quotient.
  4. If possible, factor the quadratic.
  5. Write the polynomial as the product of factors.

Example 2: Using the Factor Theorem to Solve a Polynomial Equation

Show that

(x+2)\left(x+2\right)
 is a factor of
x36x2x+30{x}^{3}-6{x}^{2}-x+30
. Find the remaining factors. Use the factors to determine the zeros of the polynomial.

Solutions

We can use synthetic division to show that

(x+2)\left(x+2\right)
is a factor of the polynomial.

Synthetic division with divisor -2 and quotient {1, 6, -1, 30}. Solution is {1, -8, 15, 0}

The remainder is zero, so

(x+2)\left(x+2\right)
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)(x28x+15)\left(x+2\right)\left({x}^{2}-8x+15\right)

We can factor the quadratic factor to write the polynomial as

(x+2)(x3)(x5)\left(x+2\right)\left(x - 3\right)\left(x - 5\right)

By the Factor Theorem, the zeros of

x36x2x+30{x}^{3}-6{x}^{2}-x+30
are –2, 3, and 5.

Try It 2

Use the Factor Theorem to find the zeros of

f(x)=x3+4x24x16f\left(x\right)={x}^{3}+4{x}^{2}-4x - 16
 given that
(x2)\left(x - 2\right)
 is a factor of the polynomial.

Solution

Licenses and Attributions

More Study Resources for You

The materials found on Course Hero are not endorsed, affiliated or sponsored by the authors of the above study guide

Show More