Name
Algebra 1 Monomials/Polynomials/Factoring packet
1
Multiplying and Dividing Monomials
Multiplying:
32 = 3  3 = 9
56=15,625
4  4 = 42 = 16
43 = (4) (4) (4) = 64
(5)(5)(5)(5)(5)(5) =
The same goes for variables:
x  x = x2
x2  x3 = (x)(x)  (x)(x)(x) = x5
(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 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: (3xy5)(4x2y3)
(3xy5)(4x2y3) = (3)(4)(x)( x2)(y5)(y3) = 12 [x(1+2) ][y(5+3)] = 12x3y8
Dividing:
4
2
6 /6 = (6)(6)(6)(6)
(6)(6)
x3/x = (x)(x)(x)
(x)
 cancel  (6)(6) (6)(6) = (6)(6) = 62= 36
(6)(6)
 cancel  (x)(x) (x) = (x)(x) = x2
(x)
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: x3/x = x3-1 = x2
Simplify: (12xy5)/(4xy3) =
12/4 = 3
x1-1 = x0
What is x0 equal to? :
Final answer =
y5-3 = y2
Any number or variable with an exponent of 0 = ?
Name
Algebra 1 Monomials/Polynomials/Factoring packet
2
Do all examples in NB.
Show all steps!
1) Multiply:
a) (5x3y2z11)(12 xy7z-4)
b) (9x5y2z4)3
c) (4x3y7z6)4(3x8y -5z -12)2
2) Multiply:
a) (6x3y2z-12)(11x5y-3 z7)
b) (8x5y-2z4)4
c) (3x6y5z8)3(5x-9y 5z -15)2
3) Divide:
a) 27x3y2z5 .
9x3y5z4
b) (4x4y5z)3
16x4y13z4
c) (2x5yz6)5
(4x11y5z14)2
4) Divide:
a) 45x3y9z5 .
18x6y5z
b) 24x8y12z9
72x10y12z8
c) 32x5y12z28
8x7y-12z14
5) (3x5y8z5)5
(9x14y20z12)2
6) (6x5y4z6)3
(12x7y8z-9)2
7) 5a(8a2 – 6a + 3) – 3a(11a2 – 10a – 5)
8) 8b(7b2 – 4b + 2) – 5(6b2 + 3b – 1)
9) 7x(4x2 - 11x + 3) - 4x(8x2 -18x + 5)
10) 5x(7x2 - 6x + 4) - 3x(10x2 -7x - 1)
11) 6y2(5y3 – 4y2 + 8y – 7) – 8y(3y3 + 6y2 – 5y – 9)
Name
Algebra 1 Monomials/Polynomials/Factoring packet
3
When MULTIPLYING monomials you
and
the exponents.
When DIVIDING monomials you
the exponents.
the coefficients
the coefficients and
1) (3x9y)(6x11y4)
2) 36x9y6z5 _
12x-9y6z4
3) (7x2yz3)3
4) 45x4y3z7 _
18x6y-3z5
5) (4x5yz3)3
(2x3y6z-2)5
6) (5x2y2z-4)(2x-5y3z)3
7) (6x7y4z3)2(2x-5y3z)3
8) (9x2y5z-11)2 _
(3x-2y2z4)5
9)
10) 4x(9x2 - 15x - 12) - 12x(3x2 + 5x - 4)
(6x2y5z3)2 _
(2x-3y2z2)5
11) 3y2(5y3 – 4y2 + 8y – 7) – 7y(3y3 + 6y2 – 5y – 9)
Name
Algebra 1 Monomials/Polynomials/Factoring packet
4
Multiplying binomials:
We have a special way of remembering how to multiply binomials called FOIL:
F:
first x  x = x2
(x + 7)(x + 5)
O:
outer x  5 = 5x
I:
inner 7  x = 7x
x2 + 5x +7x + 35 (then simplify)
L:
last
7  5 = 35
x2 + 12x + 35
1) (x - 5)(x + 4)
2) (x - 6)(x - 3)
3) (x + 4)(x + 7)
4) (x + 3)(x - 7)
5) (3x - 5)(2x + 8)
6) (11x - 7)(5x + 3)
7) (4x - 9)(9x + 4)
8)(x - 2)(x + 2)
9) (x - 2)(x - 2)
10) (x - 2)2
11) (5x - 4) 2
12) (3x + 2)2
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) 9x4 + 3x3 + 12x2
GCF: coefficients: 3
Variable (x) : x2
GCF: 3x2
What’s left? Division of monomials:
9x4/3x2
3x3 /3x2
12x2/3x2
3x2
x
4
Factored Completely:
3x2 (3x2 + x+ 4)
Factor each problem using the GCF and check by distributing:
9
7
5
4
3 2
2 3
4
1) 14x - 7x + 21x
2) 26x y - 39x y + 52x y - 13xy
6
5
11
10
4
5 2
3) 32x - 12x - 16x
9
8
5
5) 24b + 4b -6b + 2b
3 3
2 2
4 4 4
2 4
5
3 3
5
6) 96a b + 48a b - 144ab
5
4
8) 75x + 15x -25x3
7) 11x y + 121x y - 88xy
5 4 3
4 3
4) 16x y - 8x y + 24x y - 32xy
3 4 5
9) 132a b c - 48a b c + 72a b c
5
5
10) 16x + 12xy - 9y
Name
Algebra 1 Monomials/Polynomials/Factoring packet
5
HOW TO FACTOR TRINOMIALS
Remember your hints:
A. When the last sign is addition
x2 - 5x + 6
1)Both signs the same
2) Both minus (1st sign)
B. When the last sign is subtraction
x2 + 5x – 36 1) signs are different
(x - )(x - )
(x - )(x + )
3) Factors of 6 w/ a sum
of 5. (3 and 2)
(x - 3)(x - 2)
2) Factors of 36 w/ a
difference of 5 (9
and 4)
3) Bigger # goes 1st sign, +
(x - 4)(x + 9)
FOIL Check!!!!!
Factor each trinomial into two binomials check by using FOIL:
1) x2 + 7x + 6
2) t2 – 8t + 12
3) g2 – 10g + 16
2
2
4) r + 4r - 21
5) d – 8d - 33
6) b2 + 5b - 6
7) m2 + 16m + 64
8) z2 + 11z - 26
9) f2 – 12f + 27
2
2
10) x - 17x + 72
11) y + 6y - 72
12) c2 + 5c - 66
13) z2 – 17z + 52
14) q2 – 22q + 121
15) w2 + 8w + 16
2
2
16) u + 6u - 7
17) j – 11j - 42
18) n2 + 24n + 144
19) t2 + 2t -35
20) d2 – 5d - 66
21) r2 – 14r + 48
2
2
22) p + p - 42
23) s + s - 56
24) b2 – 14b + 45
25) f2 + 15f + 36
26) n2 + 7n - 18
27) z2 + 10z - 24
2
2
28) h + 13h + 24
29) w + 29w + 28
30) v2 – 3v – 18
31) y2 - 9
32) g2 – 36
33) t2 – 121
2
2
34) 9k – 25
35) 144m – 49
36) 64e2 – 81
37) a2 + 100
38) w2 – 44
39) d2 – d – 9
Factor using GCF and then factor the trinomial (then check):
40) 4b2 + 20b + 24
41) 10t2 – 80t + 150
42) 9r2 + 90r - 99
43) 3g3 + 27g2 + 60g
44) 12x6 + 72x5 + 60x4
45) 8c9 + 40c8 - 192c7
2
2
46) 12d – 12
47) 25r – 100
48) 5z5 – 320z3
Name
Algebra 1 Monomials/Polynomials/Factoring packet
Case II Factoring
Factoring a trinomial with a coefficient for x2 other than 1
Factor:
6x2 + 5x – 4
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):
6x2 + 5x – 4
6·4 = 24
* Now find factors of 24 with a difference of 5
8 and 3 [with the 8 going to the + (+5x)]
6x2 + 8x – 3x - 4
3) SPLIT THE MIDDLE and reduce each side:
6x2 + 8x | – 3x – 4
Take Out: 2x and -1
(3x + 4) (3x + 4)
*When you’re done the binomial on each side
should be the same.
4) Your binomial factors are (2x -1) and (3x + 4)
5) FOIL CHECK
(2x – 1)(3x + 4)
6x2 –8x + 3x – 4
6x2 + 5x – 4
Extra Problems: (Remember... GCF 1st)
1) 7x2 + 19x – 6
2) 36x2 - 21x + 3
3) 12x2 - 16x + 5
4) 20x2 +42x – 20
5) 9x2 - 3x – 42
6) 16x2 - 10x + 1
7) 24x2 + x – 3
8) 9x2 + 35x – 4
9) 16x2 + 8x + 1
10) 48x2 + 16x – 20
6
Name
Algebra 1 Monomials/Polynomials/Factoring packet
7
Factor each trinomial and FOIL Check:
1) x2 – 6x – 72
2) x2 + 14x + 13
3) x2 – 19x + 88
4) x2 + 2x – 63
5) x2 – 196
6) x2 – 1
7) x2 + 20x + 64
8) x2 + 11x - 12
9) x2 - 12x + 35
10) x2 - 17x + 70
11) x2 + 14x - 72
12) x2 + 5x – 36
13) x2 - 20x + 96
14) x2 - 24x + 144
15) x2 + 10x + 25
Factor using the GCF:
10
9
8
16) 24x - 144x + 48x
17) 64x5y3 – 40x4y4 + 32x3y4 – 8x2y3
Factor using the GCF and then factor the quadratic:
18) x4 – 15x3 + 56x2
19) 4x2 + 24x – 240
20) 5x3 – 5x2 –360x
21) 12x2 – 243
22) 16x2 – 16
23) 8x17 – 512x15
Mixed Problems:
24) 49x2 – 25
25) 4x2 – 121
26) x4 – 36
27) x16 – 64
28) x100 – 169
29) 48x8 – 12
30) 25x2 – 100
31) 36x4 – 9
32) 100x2 – 225
33) x2 + 64
34) x2 – 48
35) x2 – 2x + 24
36) x2 + 11x – 30
37) 5x2 + 20
38) 7x2 – 7x - 84
Name
Algebra 1 Monomials/Polynomials/Factoring packet
8
1-Step Factoring: Factor each quadratic. If the quadratic is unable to be
factored, your answer should be PRIME.
Examples:
(last sign +)
(last sign - )
x2 – 10x + 24
x2 + x – 12
Same sign, both Different Signs
Factors of 24, sum of 10
Factors of 12, diff. of 1
(x – 6)(x – 4)
(x + 4)(x – 3)
1) x2 + 5x + 4
4) g2 + 5g – 50
7) s2 – 9s + 20
10) x2 – 6x – 7
13) g2 – 5g – 84
16) p2 – 81
19) z2 + 9z – 36
22) b2 – 5b – 36
25) y2 – 4y – 60
28) x2 + 61x + 60
31) a2 + 4a – 96
34) t2 + 21t + 108
2) a2 – 12a + 35
5) t2 – 2t + 48
8) j2 + 7j + 12
11) n2 -25
14) z2 + 17z + 72
17) w2 – w – 132
20) h2 + 12h + 36
23) x2 – 36
26) v2 + 16v – 60
29) g2 – 23g + 60
32) y2 – y – 110
35) w2 – 64
(D.O.T.S)
x2 – 49
Diff of Two Sq.
(x + 7)(x – 7)
3) f2 – 3f – 18
6) x2 – 100
9) k2 + 2k – 24
12) c2 – 13c – 40
15) q2 – 3q + 18
18) x2 + 13x – 48
21) r2 + 5r + 36
24) m2 – 20m + 36
27) r2 + 7r – 60
30) b2 – 121
33) x2 + x + 90
36) x2 – 14x + 49
2-Step Factoring: Factor using the GCF and then try to factor what’s left.
Example:
6x2 – 18x + 12
6(x2 – 3x + 2)
6(x – 2)(x – 1)
37) 5x2 + 10x - 120
40) 6d2 + 60d + 150
43) 7f2 + 84f + 252
46) 5g2 - 245
38) 3w2 -33w +90
41) 9x2 - 36
44) 2x2 – 2x - 180
47) 9k2 – 99k + 252
39) 8t2 – 32t - 256
42) 10z2 + 50z - 240
45) 4s2 - 144
48) 25k2 – 225
Case II: Factor using your steps for Case II factoring. Remember GCF is always the 1
step of any type of factoring!!!
Example:
49) 2x2 – 7x - 30
52) 18y2 + 19y + 5
55) 12s2 – 22s - 20
58) 40x2 + 205x + 25
6x2 – 5x – 4
6x2 -8x + 3x - 4
(mult. 1st by last, -24)
(think: factor using F of 24, diff= 5)
6x2 – 8x | + 3x - 4
SPLIT THE MIDDLE
2x and + 1
Reduce each side
(3x - 4) and (3x – 4) Matching binomials
(2x + 1)(3x – 4)
50) 12s2 + 19s + 4
51) 18c2 + 9c - 2
2
53) 15f – 14f + 3
54) 15k2 + 7k - 8
56) 24d2 – 6d - 30
57) 21w2 + 93w + 36
2
59) 100z + 10z - 20
60) 24r2 – 90r + 21
st
Name
Algebra 1 Monomials/Polynomials/Factoring packet
9
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:
x2 – xy – 6y2
Usually, we start off with (x + )(x - ). We do this because we know when we FOIL, with the F
(first), we need x ∙ x to get us x2. Then we follow the rest of the rules.
This one is just a little different. Now we’ll start with (x + __ y)(x - __ y), because not only do
we need the x ∙ x to get x2, we’ll need a y ∙ y to get us the y2 we need at the end of the trinomial
when we do the L (last).
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
1st sign tells us to put the bigger #.
(x + 2y)(x - 3y)
Examples:
1) x2 – 11xy + 28y2
2) c2 + 3cd – 54d2
3) s2 – 25t2
4) 3u2 + 36uv + 105v2
5) 5a2 – 25ab – 180b2
6) 49f 2 – 25g2
7) 7r2 – 175s2
8) 36x2 – 9y2
9) 225w2 – 100x2