Factoring - Factoring Special Products
Objective: Identify and factor special products including a difference ofsquares, perfect squares, and sum
and difference of cubes.
When factoring there are a few special products that, if we can recognize them,can help us factor
polynomials. The first is one we have seen before. When multiplying special products we found that a
sum and a difference could multiply to a difference of squares. Here we will use this special product to
help us factor
Difference of Squares: a2 − b2 = (a + b)(a − b)
If we are subtracting two perfect squares then it will always factor to the sum and difference of the
square roots.
Example 1.
x2 − 16 Subtracting two perfect squares,the square roots are x and 4
(x + 4)(x − 4) Our Solution
Example 2.
9a2 − 25b2Subtracting two perfect squares,the square roots are 3a and 5b
(3a +5b)(3a − 5b) Our Solution
It is important to note, that a sum of squares will never factor. It is alwaysprime. This can be seen if we
try to use the ac method to factor x2 + 36.
Example 3.
x2 + 36
No bx term, we use 0x.
x2 + 0x + 36
Multiply to 36, add to 0
1 · 36, 2 · 18, 3 · 12, 4 · 9, 6 · 6
No combinations that multiply to 36 add to 0Prime, cannot
factor Our Solution
It turns out that a sum of squares is always prime.
Sum of Squares: a2 + b2 = Prime
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Example 4.
a4 − b4
Difference of squares with roots a2 and b2
(a2 + b2)(a2 − b2)
The first factor is prime,the second is a difference of squares!
(a2 + b2)(a + b)(a − b)
Our Solution
Example 5.
x4 − 16
Difference of squares with roots x2and 4
(x2 + 4)(x2 − 4)
The first factor is prime,the second is a difference of squares!
(x2 + 4)(x +2)(x − 2)
Our
SolutionExample 6.
x2 − 6x + 9
Multiply to 9, add to − 6
The numbers are − 3 and − 3,the same! Perfect square
(x − 3)2
Use square roots from first and last terms and sign from the middle
Example 7.
4x2 + 20xy + 25y2
Multiply to 100, add to 20
The numbers are 10 and 10,the same!Perfect square
(2x + 5y)2
Use square rootsfrom firstandlast terms and signfrom themiddle
Example 8.
m3 − 27
We have cube rootsmand 3
(m 3)(m23m 9)
Use formula, use SOAP to fill in signs
(m − 3)(m2 + 3m+9)
Our Solution
Example 9.
125p3 +8r3
We have cube roots 5p and 2r
(5p 2r)(25p210r 4r2)
Use formula, use SOAP to fill in signs
(5p + 2r)(25p2 − 10r +4r2)
Our Solution
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Exponent Laws
The exponent laws, also called the laws of indices (Higgens 1998) or power rules (Derbyshire
2004, p. 65), are the rules governing the combination of
exponents
(
powers
).The laws are given
by
(1)
(2)
(3)
(4)
(5)
(6)
(7)
where quantities in the
denominator
are taken to be nonzero. Special cases include
(8)
and
(9)
for
. The definition
is sometimes used to simplify formulas, but it should be kept in
mind that this equality is a definition and not a fundamental mathematical truth (Knuth 1992;
Knuth 1997, p. 56).
Note that these rules apply in general only to
real
quantities, and can give
manifestly
wrong
results if they are blindly applied to complex quantities. For example,
(10)
In particular, for complex
and real
,
(11)
where
is the
complex argument
.
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MULTIPLICATION OF POLYNOMIALS
The general rule is that
each term in the first factor has to multiply each term in the other
factor