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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)
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