When dealing with "factoring", we have encountered many different "patterns" (or "structures") that were used to arrive at the factors. Keeping these patterns (or structures) in mind may prove to be beneficial when working with higher level polynomials.
Let's take a look at a few factoring "patterns" that we want to remember.
Substitution may be used to expand the use of the pattern,
along with your knowledge of working with exponents.
Pattern:
Difference of Perfect Squares
Factor: x2 - 36
(x - 6) (x + 6)
Even
powers are perfect squares.
|
Factor: x4 - 16
Re-write to show the perfect squares.
(x2 )2 - 42
Apply the pattern involved with the difference of perfect squares.
(x2 - 4) (x2 + 4)
Continue factoring:
(x - 2) (x + 2) (x2 + 4)
|
Factor: x8 - 256
Again, perfect squares on both sides.
(x4 )2 - 162
Use the same pattern.
(x4 - 16) (x4 + 16)
(x2 - 4) (x2 + 4) (x4 + 16)
(x - 2)(x + 2)(x2 + 4)(x2 + 16) |
Pattern:
Quadratic Trimonial Factoring
Factor: x2 + 3x - 10
(x - 2) (x + 5) |
Factor: x4 + 3x2 - 10
This is the same "quadratic pattern" we listed, but x2 has replaced x.
(x2 )2 - 3(x2) - 10
So replace x with x2 in the factors.
(x2 - 2) (x2 + 5)
|
Factor: x4 - 12x2 + 27
This is the same "quadratic pattern" as:
x2 -12x + 27, so replace x with
x2 in the factors.
(x2 - 9) (x2 - 3)
Continue factoring:
(x - 3)(x + 3)(x2 - 3)
|
Pattern:
Common Factors
Factor: 3x3 - 2x2 - 2x.
x (3x2 - 2x - 2)
quadratic pattern
x (3x + 1) (x - 2)
|
Factor: 3x6 - 2x5 - 2x4
Use the same approach,
but factor out a larger power.
x4 (3x2 - 2x - 2)
quadratic pattern
x4 (3x + 1) (x - 2)
|
Factor: 4x9 - 26x8 + 30x7
Same approach with
larger common factor.
2x7 ( 2x2 - 13x + 15 )
quadratic pattern
2x7 ( 2x - 3) (x - 5) |
These "patterns" and others will be put to use in the next lesson on
Factoring Higher Power Polynomials.
