Sat math Essentials

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A L G E B R A R E V I E W
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7 3
Three Kinds of Polynomials
■
A 
monomial
is a polynomial with one term, such as 5
b
6
.
■
A 
binomial
is a polynomial with two unlike terms, such as 2
x
+ 4
y
.
■
A 
trinomial
is a polynomial with three unlike terms, such as 
y
3
+ 8
z
2.

---

Now combine like terms:
14
x
9
y
18
z
To multiply monomials, multiply their coefficients and multiply like variables by adding their exponents.
Example
(
4
a
3
b
)(6
a
2
b
3
) 
(
4)(6)(
a
3
)(
a
2
)(
b
)(
b
3
) 
24
a
5
b
4
To divide monomials, divide their coefficients and divide like variables by subtracting their exponents.
Example
1
1
0
5
x
x
5
4
y
y
7
2
(
1
1
0
5
)(
x
x
5
4
)(
y
y
7
2
) 
2
x
3
y
5
To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the
products.
Example
8
x
(12
x
3
y
9)
Distribute.
(8
x
)(12
x
) 
(8
x
) (3
y
) 
(8
x
)(9)
Simplify.
96
x
2
24
xy
72
x
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and add the quotients.
Example
6
x
1
6
8
y
42
6
6
x
1
6
8
y
4
6
2
x
3
y
7
Practice Question
Which of the following is the solution to 
1
2
8
4
x
x
8
3
y
y
5
4
?
a.
4
x
3
5
y
b.
18
2
x
1
4
1
y
9
c.
42
x
11
y
9
d.
3
x
4
5
y
e.
x
6
5
y
Answer
d.
To find the quotient:
1
2
8
4
x
x
8
3
y
y
5
4
Divide the coefficients and subtract the exponents.
3
x
8
4
3
y
5
4
3
x
4
5
y
1
3
x
4
5
y
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A L G E B R A R E V I E W
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7 4

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FOIL
The FOIL method is used when multiplying binomials. FOIL represents the order used to multiply the terms:
F
irst,
O
uter,
I
nner, and 
L
ast. To multiply binomials, you multiply according to the FOIL order and then add the
products.
Example
(4
x
2)(9
x
8)
F
: 4
x
and 9
x
are the 
first
pair of terms.
O
: 4
x
and 8 are the 
outer
pair of terms.
I
: 2 and 9
x
are the 
inner
pair of terms.
L
: 2 and 8 are the 
last
pair of terms.
Multiply according to FOIL:
(4
x
)(9
x
) 
(4
x
)(8) 
(2)(9
x
) 
(2)(8) 
36
x
2
32
x
18
x
16 
Now combine like terms:
36
x
2
50
x
16
Practice Question
Which of the following is the product of 7
x
3 and 5
x
2?
a.
12
x
2
6
x
1
b.
35
x
2
29
x
6
c.
35
x
2
x
6
d.
35
x
2
x
6
e.
35
x
2
11
x
6
Answer
c.
To find the product, follow the FOIL method:
(7
x
3)(5
x
2)
F
: 7
x
and 5
x
are the 
first
pair of terms.
O
: 7
x
and 
2 are the 
outer
pair of terms.
I
: 3 and 5
x
are the 
inner
pair of terms.
L
: 3 and 
2 are the 
last
pair of terms.
Now multiply according to FOIL:
(7
x
)(5
x
) 
(7
x
)(
2) 
(3)(5
x
) 
(3)(
2) 
35
x
2
14
x
15
x
6 
Now combine like terms:
35
x
2
x
6
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A L G E B R A R E V I E W
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7 5

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Factoring
Factoring
is the reverse of multiplication. When multiplying, you find the product of factors. When factoring,
you find the factors of a product.
Multiplication: 3(
x
y
) 
3
x
3
y
Factoring: 3
x
3
y
3(
x
y
)
Three Basic Types of Factoring
■
Factoring out a common monomial:
18
x
2
9
x
9
x
(2
x
1)
ab
cb
b
(
a
c
)
■
Factoring a quadratic trinomial using FOIL in reverse:
x
2
x
20 
(
x
4) (
x
4)
x
2
6
x
9 
(
x
3)(
x
3) 
(
x
3)
2
■
Factoring the difference between two perfect squares using the rule 
a
2
b
2
(
a
b
)(
a
b
):
x
2
81 
(
x
9)(
x
9)
x
2
49 
(
x
7)(
x
7)
Practice Question
Which of the following expressions can be factored using the rule 
a
2
b
2
(
a
b
)(
a
b
) where 
b
is an
integer?
a.
x
2
27
b.
x
2
40
c.
x
2
48
d.
x
2
64
e.
x
2
72
Answer
d.
The rule 
a
2
b
2
(
a
b
)(
a
b
) applies to only the difference between perfect squares. 27, 40, 48,
and 72 are not perfect squares. 64 is a perfect square, so 
x
2
64 can be factored as (
x
8)(
x
8).
Using Common Factors
With some polynomials, you can determine a 
common factor
for each term. For example, 4
x
is a common fac-
tor of all three terms in the polynomial 16
x
4
8
x
2
24
x
because it can divide evenly into each of them. To fac-
tor a polynomial with terms that have common factors, you can divide the polynomial by the known factor to
determine the second factor.
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A L G E B R A R E V I E W
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7 6

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Example
In the binomial 64
x
3
24
x
, 8
x
is the greatest common factor of both terms.
Therefore, you can divide 64
x
3
24
x
by 8
x
to find the other factor.
64
x
3
8
x
24
x
6
8
4
x
x
3
2
8
4
x
x
8
x
2
3
Thus, factoring 64
x
3
24
x
results in 8
x
(8
x
2
3).
Practice Question
Which of the following are the factors of 56
a
5
21
a
?
a.
7
a
(8
a
4
3
a
)
b.
7
a
(8
a
4
3)
c.
3
a
(18
a
4
7)
d.
21
a
(56
a
4
1)
e.
7
a
(8
a
5
3
a
)
Answer
b.
To find the factors, determine a common factor for each term of 56
a
5
21
a.
Both coefficients (56 and
21) can be divided by 7 and both variables can be divided by 
a.
Therefore, a common factor is 7
a.
Now,
to find the second factor, divide the polynomial by the first factor:
56
a
5
7
a
21
a
8
a
5
a
1
3
a
Subtract exponents when dividing.
8
a
5 
1
3
a
1 
1
8
a
4
3
a
0
A base with an exponent of 0 
1.
8
a
4
3(1)
8
a
4
3
Therefore, the factors of 56
a
5
21
a
are 7
a
(8
a
4
3).
Isolating Variables Using Fractions
It may be necessary to use factoring in order to isolate a variable in an equation.
Example
If
ax
c
bx
d
, what is 
x
in terms of
a
,
b
,
c
, and 
d
?
First isolate the 
x
terms on the same side of the equation:
ax
bx
c
d
Now factor out the common 
x
term:
x
(
a
b
) 
c
d
Then divide both sides by 
a
b
to isolate the variable 
x
:
x
(
a
a
b
b
)
a
c
d
b
Simplify:
x
a
c
d
b
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A L G E B R A R E V I E W
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7 7

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Practice Question
If
bx
3
c
6
a
dx,
what does 
x
equal in terms of
a, b, c,
and 
d?
a.
b
d
b.
6
a
5
c
b
d
c.
(6
a
5
c
)(
b
d
)
d.
6
a
b
d
5
c
e.
6
b
a
d
5
c
Answer
e.
Use factoring to isolate 
x
:
bx
5
c
6
a
dx
First isolate the 
x
terms on the same side.
bx
5
c
dx
6
a
dx
dx
bx
5
c
dx
6
a
bx
5
c
dx
5
c
6
a
5
c
Finish isolating the 
x
terms on the same side.
bx
dx
6
a
5
c
Now factor out the common 
x
term.
x
(
b
d
) 
6
a
5
c
Now divide to isolate 
x.
x
(
b
b
d
d
)
6
b
a
d
5
c
x
6
b
a
d
5
c
Q u a d r a t i c Tr i n o m i a l s
A 
quadratic trinomial
contains an 
x
2
term as well as an 
x
term. For example,
x
2
6
x
8 is a quadratic trino-
mial. You can factor quadratic trinomials by using the FOIL method in reverse.
Example
Let’s factor 
x
2
6
x
8.
Start by looking at the last term in the trinomial: 8. Ask yourself, “What two integers, when multiplied together,
have a product of positive 8?” Make a mental list of these integers:
1 
8
1 
8
2 
4
2 
4
Next look at the middle term of the trinomial:
6
x
. Choose the two factors from the above list that also add
up to the coefficient 
6:
2 and 
4
Now write the factors using 
2 and 
4:
(
x
2)(
x
4)
Use the FOIL method to double-check your answer:
(
x
2)(
x
4) 
x
2
6
x
8
The answer is correct.