Learn how to factor a higher-degree polynomial by using a quadratic pattern with these carefully chosen examples.
Example #1
Factor x^{4} + 7x^{2} + 6 by using a quadratic pattern
Step 1
Write x^{4} + 7x^{2} + 6 in the pattern of a quadratic expression so you can factor it like one by making a temporary substitution of variables.
Let y = x^{2} and substitute y for x^{2}
x^{4} + 7x^{2} + 6 = (x^{2})^{2} + 7(x^{2}) + 6
x^{4} + 7x^{2} + 6 = (y)^{2} + 7(y) + 6
Step 2
Factor y^{2} + 7y + 6
y^{2} + 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.
y^{2} + 7y + 6 = (y + 1 )(y + 6)
Step 3
Substitute back to the original variable
(y + 1 )(y + 6) = (x^{2} + 1)(x^{2} + 6)
Example #2
Factor x^{4} - 4x^{2} - 45 by using a quadratic pattern
Step 1
Write x^{4} - 4x^{2} - 45 in the pattern of a quadratic expression so you can factor it like one by making a temporary substitution of variables.
Let y = x^{2} and substitute y for x^{2}
x^{4} - 4x^{2} - 45 = (x^{2})^{2} - 4(x^{2}) - 45
x^{4} - 4x^{2} - 45 = (y)^{2} - 4(y) - 45
Step 2
Factor y^{2} - 4y - 45
y^{2} - 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.
y^{2} - 4y - 45 = (y - 9)(y + 5)
Step 3
Substitute back to the original variable
(y - 9)(y + 5) = (x^{2} - 9)(x^{2} + 5)
Factor completely
(y - 9)(y + 5) = (x^{2} - 9)(x^{2} + 5) = (x - 3)(x + 3)(x^{2} + 5)
Jan 26, 23 11:44 AM
Jan 25, 23 05:54 AM