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WHAT TO DO: |
HOW TO DO IT: |
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1. Factor out all common factors:
Distributed left/right: |
1. ax + ay = a(x + y)
ax + bx = (a + b)x |
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2. Perfect Square Trinomial:
Binomial Squared: |
2. a2x2 ± 2abx + b2
= (ax ± b)2 |
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3. Difference of Squares:
Conjugate Pairs |
3. A2 − B2 = (A + B)(A − B)
a2x2 − b2y2 = (ax + by)(ax − by)
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| 4. Difference of Like-Even Powers: n = 2m
Factor as the difference of squares.
Repeat as long as the binomials are
factorable. |
4. xn − yn = x2m − y2m
= (xm + ym)(xm − ym)
{ m = 2p, (xm + ym) is prime.}
xn − yn = (xm + ym) (x p + y p)(x p − y p)
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5. Factor by grouping. Polynomials with four
or more terms may be rearranged and grouped
to a recognizable form:
i) ax + by − bx − ay
ii) x2 − y2 − 4y − 4
iii) x2 +2xy + y2 + 4x + 4y + 4 |
i) ax + by − bx − ay = ax − bx − ay + by
= x(a − b) − y(a − b) = (x − y)(a − b)
ii) x2 − y2 − 4y − 4 = x2 − (y2 + 4y + 4)
x2 − (y + 2)2
= (x + y + 2)(x − y − 2)
iii) x2 +2xy + y2 + 4x + 4y + 4 = (x + y)2
+4(x + y) + 4
= (x + y + 2) 2
or [(x + y) + 2] 2 = (x + y + 2) 2 |
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6. General Rule for factoring the sum and difference of the same odd power: n = (2m + 1)
xn
yn = (x
y)(xn-1 ± xn-2
y + xn-3 y2 ± xn-4 y3 +
.. + x2 yn-3 ± x yn-2 + yn-1)
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