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Study Guides > College Algebra

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

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