Chapter Twelve — Factoring Polynomials
The Humongous Book of Algebra Problems
263
Write each term as a product of the greatest common factor and the quotients
calculated above.
Rewrite the expression as the greatest common factor 9xy
2
times the sum of the
remaining factors of each term.
18xy
4
+ 27x
3
y
2
= 9xy
2
(2y
2
+ 3x
2
)
Verify that 9xy
2
(2y
2
+ 3x
2
) is correct by applying the distributive property.
The result is equal to the original expression, so the factored form of the
expression is correct.
Note: Problems 12.15–12.16 present the steps used to factor the expression
40x
3
y
5
z
7
+ 12x
2
y
3
z
3
+ 36x
4
yz
6
.
12.15 Identify the greatest common factor of 40x
3
y
5
z
7
, 12x
2
y
3
z
3
, and 36x
4
yz
6
.
Generate the prime factorizations of each term.
All three terms have four factors in common: 2, x, y, and z. Raise each common
factor to its lowest power in all three factorizations to calculate the greatest
common factor (GCF).
GCF: 2
2
· x
2
· y
1
· z
3
= 4x
2
yz
3
A number
raised to the
zero power equals
one, so x
0
= y
0
= 1.
In other words,
pull 9xy
2
out of each
term. You’re left with
2y
2
+ 3x
2
. Put those
leftovers in parentheses
and multiply by the
greatest common
factor:
9xy
2
(2y
2
+ 3x
2
).
12x
2
y
3
z
3
and 36x
4
yz
6
both contain the
prime factor 3, but
40x
3
y
5
z
7
does not, so 3
doesn’t appear in the
greatest common
factor.
Here are all the
powers of the common
factors, with the lowest of
each circled:
Chapter Twelve — Factoring Polynomials
The Humongous Book of Algebra Problems
264
Note: Problems 12.15–12.16 present the steps used to factor the expression
40x
3
y
5
z
7
+ 12x
2
y
3
z
3
+ 36x
4
yz
6
.
12.16 Factor the greatest common factor out of the expression.
According to Problem 12.15, the greatest common factor is 4x
2
yz
3
. Divide this
quantity into each term of the expression.
Factor 4x
2
yz
3
out of the expression by multiplying it by the sum of the quotients
listed above.
40x
3
y
5
z
7
+ 12x
2
y
3
z
3
+ 36x
4
yz
6
= 4x
2
yz
3
(10xy
4
z
4
+ 3y
2
+ 9x
2
z
3
)
12.17 Factor the expression: 16x
2
y
3
– 12xy
2
.
Use the technique described in Problems 12.1–12.12 to calculate the greatest
common factor of the expression: 4xy
2
. Divide each term of the expression by
4xy
2
.
Factor the expression by writing it as the product of the greatest common factor
and the quotients above.
16x
2
y
3
– 12xy
2
= 4xy
2
(4xy – 3)
12.18 Factor the expression: –4x
7
y
3
– 20x
5
y
11
.
Divide each term of the expression by the greatest common factor: –4x
5
y
3
.
Factor the expression by writing it as the product of the greatest common factor
and the preceding quotients.
–4x
7
y
3
– 20x
5
y
11
= –4x
5
y
3
(x
2
+ 5y
8
)
Distribute
4xy
2
to check
your answer:
4xy
2
(4xy) + 4xy
2
(–3) =
16x
2
y
3
– 12xy
2
The greatest
common factor is
negative because
both of the terms are
negative—both are
multiplied by –1, so you
can factor that out
as well.
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