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Chapter Twelve — Factoring Polynomials
The Humongous Book of Algebra Problems
267
Factor the binomial x
2
– 2 out of the expression.
(x
2
– 2)(4x
3
– 5)
Therefore, the factored form of 4x
5
– 8x
3
– 5x
2
+ 10 is (x
2
– 2)(4x
3
– 5).
12.25 Factor the expression: 96x
4
+ 36x
3
– 160x – 60.
Before you factor by grouping, note that all of the terms share the
common factor 4. Factor it out of the expression.
4(24x
4
+ 9x
3
– 40x – 15)
The ﬁrst two terms of the quartic polynomial share common factor 3x
3
;
the last two terms share common factor –5.
Factor the common binomial out of the expression.
Common Factor Patterns
Difference of perfect squares/cubes, sum of perfect cubes
12.26 Factor the expression: a
2
b
2
.
The expression a
2
b
2
is a difference of perfect squares. In other words, the
expression consists of one squared quantity (b
2
) subtracted from another (a
2
).
All differences of perfect squares are factored using the formula
(a
2
b
2
) = (a + b)(ab).
12.27 Factor the expression: w
2
– 36.
Notice that w
2
and 36 are perfect squares, as each is equal to some quantity
times itself: w · w = w
2
and 6 · 6 = 36.
w
2
– 36 = w
2
– 6
2
According to Problem 12.26, a polynomial of the form a
2
b
2
has factors
(a + b)(ab). Here, a = w and b = 6.
In Problem
12.23, fac-
toring out a 1
didnt change
the binomial. When
you factor out a
1 in this problem, it
changes every term
in the parentheses
into its opposite.
“Quartic
means the highest
power of x in the
polynomial is 4.
Leave
this 4 (that
got factored out
in the previous step)
in front of the
expression.
A perfect
square is a
number you get by
multiplying something
times itself. That
means 4 is a perfect
square because 2 times
itself equals 4:
2
˙
2 = 2
2
= 4.
A DIFFERENCE of
perfect squares just
means subtraction
is involved.
Chapter Twelve — Factoring Polynomials
The Humongous Book of Algebra Problems
268
Therefore, the factored form of w
2
– 36 is (w + 6)(w – 6).
12.28 Factor the expression: 4x
2
– 81.
This binomial is the difference of perfect squares: (2x)(2x) = 4x
2
and
(9)(9) = 81. Therefore, it should be factored using the formula
a
2
b
2
= (a + b)(ab). If a
2
b
2
= 4x
2
– 81, then a = 2x and b = 9.
12.29 Factor the expression: 98x
2
– 200.
Both terms are divisible by 2, so factor that value out of the expression.
2(49x
2
– 100)
The resulting binomial, 49x
2
– 100, is a difference of perfect squares: (7x)
2
=
49x
2
and 10
2
= 100.
2(7x + 10)(7x – 10)
12.30 Factor the expression: 4x
4
– 64.
The terms of this expression have a greatest common factor of 4, so factor 4 out
of the polynomial.
4(x
4
– 16)
The binomial x
4
– 16 is a difference of perfect squares: (x
2
)
2
= x
4
and 4
2
= 16.
4(x
2
+ 4)(x
2
– 4)
The expression is not yet fully factored, because x
2
– 4 is, itself, a difference of
perfect squares: (x
2
– 4) = (x + 2)(x – 2). Substitute the factored form of x
2
– 4
into the polynomial.
4x
4
– 64 = 4(x
2
+ 4)(x + 2)(x – 2)
(w – 6)(w + 6) is also
correct. The order of
the factors doesnt
matter.
a” is the
thing you multiply
times itself to get
the left term. In this
case a = 2x because
(2x)
2
= 4x
2
. “b” is the
thing you multiply
times itself to get
the term that’s
subtracted.
Always
check to
see if there is a
greatest common
factor before you
try to factor an
expression any other
way. In this problem,
if you dont factor
the 2 out rst, you
dont end up with
a difference
of perfect
squares.
After factoring the
difference of perfect squares,
it turns out one of the factors is
ALSO a difference of perfect squares.
You cant factor x
2
+ 4, because
theres no formula for the SUM of
perfect squares.

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