Steps in Factoring
1) Factor out the greatest common factor
(GCF). (There will not always be one).
2) Count the number of terms.
Two terms: Look to see if you have a
difference of squares or a sum or difference of cubes.
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Difference of squares: X -Y => (X+Y)(X-Y)
X + Y => PRIME
Difference of cubes: X - Y=>(X-Y)(X +XY +Y )
Sum of cubes: X + Y => (X+Y)(X -XY + Y )
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Three terms: Look for two binomials.
A. Trial and error method:
2x - x -10
*Remember! (A-C)(B+D) = AB+AD-BC-CD.
The factors of 10 take the place of C and D.
The factors of 2x (2x and x) take the place of A and B.
Try: (2x 5)(x 2) or (2x 10)(x 1)
At this point, you must use trial and error to arrange the
factors and signs so that the original trinomial can be obtained
by combining the binomials in the FOIL manner.
(2x-5)(x+2) =2x +4x-5x-10 combine like terms to get = 2x -x-10
Clues for factoring trinomials by trial and error.
If the sign of the last term is +, the middle sign of the
binomials will have the same sign as the second term in the
Example: x -3x +2 = (x- 2)(x- 1) or x +5x+6= (x+3)(x+2)
If the sign of the last term is -, the middle sign of the
binomials will be + and-.
Example: x -5x-6= (x-6)(x+1) or x +x -56= (x+8)(x-7)
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B. Grouping number method:
3x -10x+8 1. Multiply the a
term of the coefficient x with the constant
( ax + bx + c ) c term. You are looking for two
numbers that multiply together to get 24 and add together to get
10 in this case.
3x -4x-6x+8 2. Rewrite the
problem. Replace the middle term ( bx ) with -4x and -6x.
3x -4x -6x+8 3. Solve by
a. Pull out a GCF from each group.
b. Collect like groups and combine remaining
Four terms: Factor by grouping method.
A. Grouping Method
am + an + bm + b 1. Count the terms. Group
the first two terms and the last two terms together.
(am + an) + (bm + bn)
a(m + n) + b(m + n) 2. Take out the common
factor in the first group and the common factor in the second
(m + n)(a + b) 3. Factor out the common
3) Check to be sure each factor is prime, if
not, repeat 1-3.
4) Check by multiplying the factors out to
see if you get the original polynomial.
REMEMBER!! FACTORING IS UNMULTIPLYING!!
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