Factor by using a quadratic pattern

Learn how to factor a higher-degree polynomial by using a quadratic pattern with these carefully chosen examples.

Example #1

Factor x4 + 7x2 + 6 by using a quadratic pattern

Step 1

Write x4 + 7x2 + 6 in the pattern of a quadratic expression so you can factor it like one by making a temporary substitution of variables.

Let y = x2 and substitute y for x2

x4 + 7x2 + 6 = (x2)2 + 7(x2) + 6

x4 + 7x2 + 6 = (y)2 + 7(y) + 6

Step 2

Factor y2 + 7y + 6

y2 + 7y + 6 = (y + ___ )(y + ___ )

To fill in the blank above, look for factors of 6 that will add up to 7.

6 × 1 = 6 and 6 + 1 = 7. 

Fill in the blank in the expression above with 1 and 6.

y2 + 7y + 6 = (y + 1 )(y + 6)

Step 3

Substitute back to the original variable

(y + 1 )(y + 6) = (x2 + 1)(x2 + 6)

Example #2

Factor x4 - 4x2 - 45 by using a quadratic pattern

Step 1

Write x4 - 4x2 - 45 in the pattern of a quadratic expression so you can factor it like one by making a temporary substitution of variables.

Let y = x2 and substitute y for x2

x4 - 4x2 - 45 = (x2)2 - 4(x2) - 45

x4 - 4x2 - 45 = (y)2 - 4(y) - 45

Step 2

Factor y2 - 4y - 45

y2 - 4y - 45 = (y + ___ )(y + ___ )

To fill in the blank above, look for factors of -45 that will add up to -4.

-9 × 5 = 45 and -9 + 5 = -4. 

Fill in the blank in the expression above with -9 and 5.

y2 - 4y - 45 = (y - 9)(y + 5)

Step 3

Substitute back to the original variable

(y - 9)(y + 5) = (x2 - 9)(x2 + 5)

Factor completely

(y - 9)(y + 5) = (x2 - 9)(x2 + 5) = (x - 3)(x + 3)(x2 + 5)

Recent Articles

  1. Irrational Root Theorem - Definition and Examples

    Dec 01, 21 04:17 AM

    What is the irrational root theorem? Definition, explanation, and easy to follow examples.

    Read More

Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.