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Math 142 — Rodriguez
Lehmann — 6.6
Factoring Special Binomials; A Factoring Strategy
I. Factoring Special Binomials
A. Perfect Squares
Numbers: 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, etc…
Variables: (x)2, (x2)2, (x3)2, (x4)2, (x5)2, etc…
Notice for variables?
B. Factoring the Difference of Two Squares: A 2 − B 2 = ( A − B ) ( A + B )
Examples:
x 4 − 25
4 x 2 − 25y 2
2x 4 − 18
C. Factoring the Sum of Two Squares: A2 + B2 is
Example:
x2 + 4
D. Factor completely.
x 4 −1
x 4 − 16y 4
81x 4 −1
II. Factoring Strategy
To factor completely a given polynomial follow these steps:
Step 1: Look for a GCF. If there is a GCF, factor it out. If the leading coefficient is negative,
factor out a negative GCF.
Step 2: Look at the number of terms in the polynomial. This determines the factoring
method you use.
2 terms in poly:
§ Is it a sum of perfect squares?
§
A2 + B2 then poly is prime
Is it a difference of perfect squares?
3 terms in poly:
§ Is it of the form x2 + bx + c?
A2 − B2 = ( A − B)( A + B)
x2 + bx + c = (x
)(x
)
Find 2 numbers that multiply to c and add to b.
If no such numbers exist, poly prime.
§
Is it of the form ax2 + bx + c? Use “Trial and Error” or “Factor by grouping”.
ac-method
4 terms in poly:
§ Try grouping – group in pairs; factor out the GCF of each pair and see if the
binomial left inside the parenthesis is the same. If it is, continue with the
process.
Step 3: Look at each factor. Can it be factored further? The poly is factored completely when
none of the factors can be factored further. (That is, when all the factors are prime
poly’s.)
Step 4: Check your factorization by multiplying. The product of all the factors should be the
original polynomial. For one variable polynomials you can also use graphing
calculator tables to check your answers.
1.
p 2 − 9 pt + 20t 2
2. 2x y + 24x y + 70x y
4
3 2
2 3
3. 8x − 13x + 10
2
4. 75r − 27y
2
2
5. 36x − 21x y + 3x y
4
3
2 2
6. 9x + 27x − x − 3
3
Lehmann — 6.6
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