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Factoring
Factoring out a Common Factor : The first step in factoring any polynomial
is to
look for anything that all the terms have in common and then factor it out using
the
distributive property .
Example: 20y^{2}  5y^{5} Here, the terms share the common factor 5y^{2} (i.e. 5 is the
largest
number that divides both 20 and 5, and both terms contain the variable y with 2
being
the smallest exponent ). So we factor it out: 20y^{2}  5y^{5} = 5y^{2}(4  y^{3})
Factoring by Grouping: Factoring by grouping is useful when we encounter
a polynomial
with more than 3 terms.
Example: 3x^{3} + x^{2}  18x  6
1. First, we group together terms that share a common factor. (3x^{3} + x^{2}) + (18x
 6)
The first group shares an x^{2} and the second shares a 6.
2. Factor out the common factor from each grouping. You should have left the
same
expression in each group . x^{2}(3x+1)+(6)(3x+1) Here that expression is 3x+1
3. Now factor out that expression. (3x + 1)(x^{2}  6)
Factoring Trinomials  Reverse FOIL: There two basic cases that
we’ll encounter:
1. The leading coefficient is a 1. This is the easier of the two cases: x^{2} + bx
+ c All
we need to do here is find two numbers whose product is c and sum is b
Example: x^{2}  7x + 10 = (x + △)(x + △) We need to find two numbers that
multiply to give us +10, but add to give us 7. Well, 5 and 2 do the trick. So
x^{2}  7x + 10 = (x + (2))(x + (5)) = (x  2)(x  5)
2. The leading coefficient is not a 1. Things are a little trickier here, but
not much.
Again, it’s just FOIL in reverse.
Example:
We need two numbers to fill in for the hearts that will multiply to 3. How about
3 and 1?
3y^{2} + 7y  20 = (3y +△)(1y + △)
Now we need two numbers to fill in for the triangles that will multiply to 20
AND when we do the INNERS and OUTERS we get 7y. We’ll use the GUESS
and CHECK method to find the two numbers we need.
Let’s try 10 and 2 first:
(3y  2)(y + 10) = 3y^{2} + 30y  2y  20 = 3y^{2} + 27y  20
That’s not it! Maybe 5 and 4?
(3y + 5)(y  4) = 3y^{2}  12y + 5y  20 = 3y^{2}  7y  20
Close, but the sign on the 7 is wrong . Easy to fix  just switch the signs on
the 5
and 4:
(3y  5)(y + 4) = 3y^{2} + 12y  5y  20 = 3y^{2} + 7y  20 Presto!!
Special Factorizations: Some polynomials are easy to factor because they fit
a
certain mold.
– Difference of Squares : F^{2}  L^{2} = (F + L)(F  L)
Example: 16x^{2}  9 = 42x^{2}  3^{2} = (4x)^{2}  3^{2} = (4x + 3)(4x  3)
– Perfect Squares : These are polynomials that factor into (F + L)^{2} or (F  L)^{2}
The pattern we’re looking for here is F^{2} + 2LF + L^{2} or F^{2}  2LF + L^{2}
Example: x^{2} + 6x + 9 = x^{2} + 2·3x + 3^{2} = (x + 3)^{2}
Example: y^{2}  10y + 25 = y^{2}  2·5y
+ 5^{2}
– Difference of Cubes : F^{3}  L^{3} = (F  L)(F^{2} + LF + L^{2})
Example: 2z^{3}  54 = 2(z^{3}  27) = 2(z^{3}  3^{3}) = 2(z  3)(z^{2} + 3z + 9)
– Sum of Cubes : F^{3} + L^{3} = (F + L)(F^{2}  LF + L^{2})
Example: n^{3} + 216 = n^{3} + 6^{3} = (n + 6)(n^{2}  6n + 36)
Strategy for Factoring:
1. Always factor out the largest common factor first. This will make life easier
for
any further factoring that may need to be done.
2. Look at the number of terms
– Two terms: Is it a difference of squares, difference of cubes or sum of cubes?
– Three terms: Is it a perfect square? Try reverse FOIL.
– Four or more terms: Try factoring by grouping.
3. Always make sure the polynomial is factored COMPLETELY.
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