Polynomials Factoring Cheat Sheet Page 2
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1
2
3
4Patterns for multiplying and factoring:
(a and b may be any monomials)
2
2
2
2
(binomial)
= perfect square trinomial
(a + b)
= a
+ 2ab + b
2
2
2
(a - b)
= a
- 2ab + b
2
2
(bino. sum)(bino. diff.) = diff of 2 squares
(a + b)(a – b) = a
– b
3
3
2
2
sum of two cubes
a
+ b
= (a + b)(a
–ab +b
)
3
3
2
2
difference of two cubes
a
– b
= (a – b)(a
+ ab +b
)
2
rd
Procedure for factoring 1x
+ bx + c
(“3
term” procedure)
rd
(1) list all the possible factors of the 3
term c, positive and negative
(2) select that pair of factors that also add to form middle term coefficient b
(3) the selected factors become the constants added to x in the two binomials
2
example: x
-7x + 10 factors
sum
1 10
11
-1 -10
-11
2
5
7
-2 -5
-7 <<<< sums to -7, select
2
so x
-7x + 10 = (x-2)(x-5)
(always check by multiplying your answer to see if you get the original problem)
2
“Organized trial-and-error” procedure for factoring ax
+ bx + c
(1) list all the possible pairs of factors of a, to be the coefficients of x
(2) list all the possible pairs of factors of c, to be the constants added to x terms
(3) multiply out all combinations of (1) and (2) until obtaining the right “bx”
2
2
example: 2x
+x – 3
factors of 2 (on x
)
factors of -3
1 and 2
-1 and 3
-1 and -2
1 and -3
combinations
result
2
(x-1)(2x+3)
2x
+x -3 <<<< that’s it
2
(-x -1)(-2x +3)
2x
–x -3
2
(x+1)(2x -3)
2x
–x -3
2
st
(-x+1)(-2x-3)
2x
+x -3 <<<< same as 1
one
2
(x+3)(2x-1)
2x
+5x -3
2
(-x+3)(-2x-1)
2x
-5x -3
2
(x-3)(2x+1)
2x
-5x -3
2
(-x-3)(-2x+1)
2x
+5x -3
(always check by multiplying your answer to see if you get the original problem)
2
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