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Name ______________________________________________ Date ____________ Color___________
Algebra I
Ms. Hahl
Multiplying and Dividing Monomials
Multiplying:
32 = 3  3 = 9
4  4 = 42 = 16
43 = (4) (4) (4) = 64
(5)(5)(5)(5)(5)(5) = 56=15,625
The same goes for variables:
(The only difference is you can’t simplify x2 like you did 32 = 9. You must leave it as x2.)
When multiplying monomials you must also deal with the coefficients.
Coefficients: Multiply the coefficients.
Variables:
When multiplying the variables of monomials you keep the base and add the
exponents. (Remember if there is no exponent written, the exponent is 1.)
Look at the previous example:
x1  x1 = x(1+1) = x2
Simplify:
Dividing:
=
 reduce 
=
=
 reduce 
=
=
=
=
Just like multiplying, when dividing monomials you must deal with the coefficients.
Coefficients : Divide the coefficients.
Variables: When dividing the variables of monomials you keep the base and subtract the exponents.
Look at the previous example:
Simplify:
=
12/4 = 3
What is x0 equal to? :
x1-1 = x0
y5-3 = y2
Any number or variable with an exponent of 0 = ? ______
Final answer =
2
Do all examples in NB.
Show all steps!!
1) Multiply:
a)
b)
c)
d)
e)
f)
2) Divide:
a)
b)
c)
d)
e)
f)
g)
h)
3) Multiply & Simplify:
a)
b)
c)
d)
e)
3
When MULTIPLYING monomials you
the exponents.
the coefficients and
When DIVIDING monomials you
_______________ the exponents.
the coefficients and
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
)
4
Multiplying binomials:
We have a special way of remembering how to multiply binomials called FOIL:
F:
O:
I:
L:
first
outer
inner
last
(then simplify)
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
Factoring using GCF:
Take the greatest common factor (GCF) for the numerical coefficient. When choosing the GCF for the
variables, if all the terms have a common variable, take the one with the lowest exponent.
ie)
GCF: coefficients:
Variable (x) :
GCF:
What’s left? Division of monomials:
/
/
Factored Completely:
Factor each problem using the GCF and check by distributing:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
/
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HOW TO FACTOR TRINOMIALS
Remember your hints:
A. When the last sign is addition
1)Both signs the same
2) Both minus (1st sign)
B. When the last sign is subtraction
1) signs are different
3) Factors of 6 w/ a sum
of 5. (3 and 2)
2) Factors of 36 w/ a difference
of 5 (9 and 4)
3) Bigger # goes 1st sign, +
FOIL to check!!!!!
Factor each trinomial into two binomials check by using FOIL:
1)
4)
7)
10)
13)
16)
19)
22)
25)
28)
31)
34)
37)
2)
5)
8)
11)
14)
17)
20)
23)
26)
29)
32)
35)
38)
3)
6)
9)
12)
15)
18)
21)
24)
27)
30)
33)
36)
39)
Factor using GCF and then factor the trinomial (then check):
40)
43)
46)
41)
44)
47)
42)
45)
48)
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Case II Factoring
Factoring a trinomial with a coefficient for x2 other than 1
Factor:
1) Look for a GCF:
a. There is no GCF for this trinomial
b. The only way this method works is if you take out the GCF (if there is one.)
2) Take the coefficient for x2 (6) and multiply it with the last term (4):
3) Factor the new trinomial:
4) Take the coefficient that you multiplied in the beginning (6) and put it back in the
parenthesis (only with the x):
5) Find the GCF on each factor (on each set of parenthesis):


6) Keep the factors left in the parenthesis:
7) FOIL CHECK
Extra Problems: (Remember... GCF 1st)
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
7
A) Factor each trinomial and FOIL Check:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
B) Factor using the GCF:
16)
17)
C) Factor using the GCF and then factor the quadratic:
18
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
38)
D) Mixed Problems:
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1-Step Factoring: Factor each quadratic. If the quadratic is unable to be factored, your answer
should be PRIME.
Examples:
(last sign +)
Same sign, both Factors of 24, sum of 10
1)
4)
7)
10)
13)
16)
19)
22)
25)
28)
31)
34)
2)
5)
8)
11)
14)
17)
20)
23)
26)
29)
32)
35)
(last sign - )
(D.O.T.S)
Different Signs
Factors of 12, diff. of 1
Diff of Two Sq.
3)
6)
9)
12)
15)
18)
21)
24)
27)
30)
33)
36)
2-Step Factoring: Factor using the GCF and then try to factor what’s left.
Example:
37)
40)
43)
46)
38)
41)
44)
47)
39)
42)
45)
48)
Case II: Factor using your steps for Case II factoring. Remember GCF is always the 1st step!!!
(mult. 1st by last)
(factor)
(put the 1st #, 6, back in)
(reduce: take 2 out of the 1st factor and 3 out of the 2nd)
Example:
49)
52)
55)
58)
50)
53)
56)
59)
51)
54)
57)
60)
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Factoring with 2 Variables
You will follow all your same factoring rules.
1) Look for a GCF.
2) Choose whether it is Case I, DOTS, or CASE II and proceed.
Remember: All the rules of factoring quadratics come from what will result when you FOIL check the
two binomials:
Example:
Usually, we start off with
We do this because we know when we FOIL, with the F (first),
we need
to get us . Then we follow the rest of the rules.
This one is just a little different. Now we’ll start with
need the
to get , we’ll need a
to get us the
we do the L (last).
, because not only do we
we need at the end of the trinomial when
Once you have that you can proceed. We will need factors of 6 with a difference of 1. The factors of 3
and 2 satisfy that and we will put the 3 with the minus sign because that is where the 1 st sign tells us to
put the bigger #.
Examples:
1)
2)
3)
4)
5)
6)
7)
8)
9)