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204
Chapter 8
8–1
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Factors and Factoring
Common Factors
Factors of an Expression
The factors of an expression are those quantities whose product is the original expression.
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Example 1: The factors of x2 9 are x 3 and x 3, because
(x 3)(x 3) x2 3x 3x 9
x2 9
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Prime Factors
Many expressions have no factors other than 1 and themselves. They are called prime expressions.
◆◆◆ Example 2: Two factors of 30 are 5 and 6. The 5 is prime because it cannot be factored.
However, the 6 is not prime because it can be factored into 2 and 3. Thus the prime factors of
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30 are 2, 3, and 5.
Example 3: The expressions x, x 3, and x2 3x 7 are all prime. They cannot be fac◆◆◆
tored further.
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Example 4: The expressions x2 9 and x2 5x 6 are not prime because they can be
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factored. They are said to be factorable.
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Factoring
Factoring is the process of finding the factors of an expression. It is the reverse of finding the
product of two or more quantities.
Multiplication
(Finding the Product)
x (x 4) x2 4x
Factoring
(Finding the Factors)
x2 4x x (x 4)
We factor an expression by recognizing the form of that expression and then applying standard rules for factoring. In the first type of factoring we will cover, we look for common factors.
Common Factors
If each term of an expression contains the same quantity (called the common factor), that quantity
may be factored out.
This is simply the distributive law,
which we studied earlier.
Common Factor
◆◆◆
ab ac a(b c)
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Example 5: In the expression
2x x2
each term contains an x as a common factor. So we write
2x x2 x (2 x)
Most of the factoring we will do will be of this type.
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Section 8–1
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(a)
(b)
(c)
(d)
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205
Common Factors
Example 6:
3x3 2x 5x4 x(3x2 2 5x3)
3xy2 9x3y 6x2y2 3xy(y 3x2 2xy)
3x3 6x2y 9x4y2 3x2(x 2y 3x2y2)
2x2 x x (2x 1)
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When factoring, don’t confuse superscripts (powers) with subscripts.
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Example 7: The expression x3 x2 is factorable.
x3 x2 x2(x 1)
But the expression
x 3 x2
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is not factorable.
Common
Error
Students are sometimes puzzled over the “1” in Example 7.
Why should it be there? After all, when you remove a chair
from a room, it is gone; there is nothing (zero) remaining
where the chair used to be. If you remove an x by factoring,
you might assume that nothing (zero) remains where the x
used to be.
2x2 x x (2x 0)?
Prove to yourself that this is not correct by multiplying
the factors to see if you get back the original expression.
Factors in the Denominator
Common factors may appear in the denominators of the terms as well as in the numerators.
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Example 8:
1
2
1p
2q
(a) 1 x x2
x
x
2
x
x
2x x 1
2
(b) p x q
y
3y y y
3
y2
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Checking
To check if factoring has been done correctly, simply multiply your factors together and see if
you get back the original expression.
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Example 9: Are 2xy and x 3 y2 the factors of
2x2y 6xy 2xy3?
Solution: Multiplying the factors, we obtain
2xy(x 3 y2) 2x2y 6xy 2xy3 (checks)
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This check will tell us whether
we have factored correctly but,
of course, not whether we have
factored completely (that is,
found the prime factors).
206
Chapter 8
◆
Factors and Factoring
CASE STUDY – CHECKING YOUR FACTORING
Here is an interesting look at mathematics: Although it may seem a little strange to have x y,
there is actually nothing wrong with it. Sometimes you will see an x changed to a y or a y changed
to an x. Normally, you can’t do that unless x y, as is the case here. Look over the steps below
and notice the factoring. Use the knowledge you gain in this chapter to check whether the factoring
is done correctly. If everything is correct, then two is the same as one and all of our mathematics
developed over the last several thousand years is wrong. Can you determine what is wrong in this
chain of math?
x2 y 2 x2 xy
(x y)(x y) x(x y)
(x y)(x y)
x(x y)
(x y)
(x y)
xyx
given x y
xxx
2x x
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Exercise 1
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Common Factors
Factor each expression and check your results.
1.
2.
3.
4.
3y2 y3
6x 3y
x5 2x4 3x3
9y 27xy
5. 3a a2 3a3
6. 8xy3 6x2y2 2x3y
7. 5(x y) 15(x y)2
a a2 a3
8. 3
4
5
3
5
2
9. x x2 x3
3ab2 6a2b 12ab
10. y
y3
y2
5m 15m2 25m3
11. 2n
8n
4n2
16y2 8y3 8y4
12. 3x
9x2
3x3
13. 5a2b 6a2c
14.
15.
16.
17.
18.
19.
a2c b2c c2d
4x2y cxy2 3xy3
4abx 6a2x2 8ax
3a3y 6a2y2 9ay3
2a2c 2a2c2 3ac
5acd 2c2d 2 bcd
20. 4b2c2 12abc 9c2
21. 8x2y2 12x2z2
Section 8–2
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207
Difference of Two Squares
22. 6xyz 12x2y2z
23. 3a2b abc abd
24. 5a3x2 5a2x3 10a2x2z
Applications
25. When a bar of length L0 is changed in temperature by an amount t, its new length L will
be L L0 L0 t, where is the coefficient of thermal expansion. Factor the right side of
this equation.
26. A sum of money a when invested for t years at an interest rate n will accumulate to an
amount y, where y a ant. Factor the right side of this equation.
27. When a resistance R1 is heated from a temperature t1 to a new temperature t, it will increase
in resistance by an amount (t t1)R1, where is the temperature coefficient of resistance. The final resistance will then be R R1 (t t1)R1. Factor the right side of this
equation.
28. An item costing P dollars is reduced in price by 15%. The resulting price C is then
C P 0.15P. Factor the right side of this equation.
29. The displacement of a uniformly accelerated body is given by
a
s v0t t2
2
Factor the right side of this equation.
30. The sum of the voltage drops across the resistors in Fig. 8–1 must equal the battery
voltage E.
Compare your result for problem
27 with Eq. A70.
E
R1
i
E iR1 iR2 iR3
Factor the right side of this equation.
31. The mass of a spherical shell having an outside radius of r2 and an inside radius r1 is
4
4
mass r23D r13D
3
3
where D is the mass density of the material. Factor the right side of this equation.
8–2
R3
FIGURE 8–1
Difference of Two Squares
Form
As we saw in Sec. 2–4, an expression of the form
a2 b2
perfect square
perfect square
minus sign
where one perfect square is subtracted from another, is called a difference of two squares. It
arises when (a b) and (a b) are multiplied together.
Difference of
Two Squares
a2 b2 (a b)(a b)
Factoring the Difference of Two Squares
Once we recognize its form, the difference of two squares is easily factored.
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R2