Pattern or Structure:
Difference of Perfect Squares
Factor: x2 - 36
(x - 6) (x + 6)
Even
powers are perfect squares.
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Factor: x4 - 16
Re-write to show perfect squares.
(x2 )2 - 42
Apply the pattern for the difference of perfect squares.
(x2 - 4) (x2 + 4)
Continue factoring:
(x - 2) (x + 2) (x2 + 4)
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Factor: x8 - 256
Again, re-write to show perfect squares.
(x4 )2 - 162
Apply the pattern.
(x4 - 16) (x4 + 16)
Repeat pattern for first factor.
(x2 - 4) (x2 + 4) (x4 + 16)
Continue factoring.
(x - 2)(x + 2)(x2 + 4)(x2 + 16) |
Pattern or Structure:
Factorable Quadratic Trimonial
Factor: x2 + 3x - 10
(x - 2) (x + 5)
Look for a possible "hidden"
(nicely factorable) quadratic in a higher power expression. |
Factor: x4 + 3x2 - 10
This is the same "quadratic pattern" we saw in the box to the left,
but x2 has replaced x.
(x2 )2 - 3(x2) - 10
So, by substitution, replace x
with x2 in the factors.
(x2 - 2) (x2 + 5)
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Factor: x4 - 12x2 + 27
This is the same "quadratic pattern" as:
x2 -12x + 27, which is nicely factorable,
so replace x with x2.
(x2 )2 - 12(x2) + 27
Factor, replacing x with with x2.
(x2 - 9) (x2 - 3)
Continue factoring:
(x - 3)(x + 3)(x2 - 3)
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Do you see how to apply this pattern of looking for a possible "hidden" quadratic? In a one variable trinomial, if the power in the middle term is half the power in the leading term, there is a "hidden" quadratic. The power values need not be both "even" numbers.
The leading power just needs to be twice as large as the middle tern power.
Consider: x10 - 12x5 + 27 = (x5 )2 - 12(x5) + 27 = (x5 - 9) (x5 - 3)
NOTE: Not all quadratic trinomials are nicely factorable over the set of integers.
Remember the quadratic formula, where trinomials can have factors that are irrational or complex values.
When factoring higher powers, this course will focus on factors over the set of integers. |
Pattern or Structure:
Common Factors
Factor: 3x3 - 2x2 - 2x.
x (3x2 - 2x - 2)
Now, apply the factorable quadratic pattern
x (3x + 1) (x - 2)
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Factor: 3x6 - 2x5 - 2x4
Use the same approach,
but factor out a larger power.
x4 (3x2 - 2x - 2)
Now, apply the factorable
quadratic pattern
x4 (3x + 1) (x - 2)
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Factor: 4x9 - 26x8 + 30x7
Same approach with
larger common factor.
2x7 ( 2x2 - 13x + 15 )
Now, apply the factorable
quadratic pattern
2x7 ( 2x - 3) (x - 5) |
