FMPCIO
Unit 2: Polynomials
Lesson 1: Multiplying & Dividing Monomials
2
y
5
3x
This is a term of a polynomial. The 3 is called the coefficIent, the x and y are called
the variables, and the 5 and 2 are the exponents. The exponents must be whole
numbers.
Types of polynomials
# of Terms
Type of Polynomial
Example
I
2
3
4
DEGREE OF POLYNOMIALS: In general, degree refers to the highest sum of
exponents of any one term of a polynomial
Example 1: State the degree of each polynomial
x 1
x+
2
+
3
a)x
—
degree:
n
m
2
-m
4
c)m
+
+
3
n
2
-
x
y
xy+1
÷
3
b)x
+
y
2
÷
degree:
degree:
Example 2: Simplify
a) (3a) (2a)
b) (12x)
)
2
) (np
3
n
2
c) (2m
d)
1 2x
5
2
3x
(3X2)
(-2x)
e)
5
12m
3
24m
)(2x
(3x
3
)
g) 2
f)
5
n
2
28m
—
I)
3
n
2
12m
3
)
5
y
(—3x
2
)
(—4xy
y
4
8x
HW:
2.1 Worksheet Attached
2.1 Multiplying & Dividing Monomials Worksheet
1! Simplify,.
a) 4
X5x
2
(7x
)
N (3aX4a)
e)
1)
y
(—7y
4
(—8
)
5
)
2! Simplify,
a) (6x)(3y)
e) (—9rXSrs
)
3
—6c4
s)
4b)
(—7b
X
3
4q)
(Sp
X
b) 2
.1) (4c4)(—I 1cr)
3+ Simplify.
a) (—4abX2b
)
2
8p
3pq
ci) 2
(—
—
5
q
)
4. Simplify.
a’ i2x
2
3x
‘
3
—24cd
e)
m
ic) 2
9m
(—6
X
3
—
(PX—60)
c)
m
(—ln)
3
(—5
)
)
(_2s12X.3st)
36,,,6
2
4rn
C,i
5
n
2
28in
g,
.
A!
54a8
2
—6a
ci’‘
—32aNø
li’
‘
3
n
2
-7rn
5. Simplify
a)
ci)
4
s
0
36r
s
2
8r
6ab
3
X
2
b
4
(3a
)
4
b
3
9a
6.
ci)
)(._5p
3
(_4pq3
2
)
b
‘
ci)
j42xy2)!
y
8
-45x
2
y
4
15x
‘
X6mn
2
n
4
e) (4m
n
6
—3rn
.
a
X2a
4
(—3
b
2
1,) )
flfl
(3m2n3X_21
3
)
2
e)
c)
b)
c)
e)
o
“.
ci) (2cjbX7a)
h) 2
Xbc
3
c
(2b
)
c) )
3ab
3
X
b
2
(7a
)
a
f) 2
(—3ab)(2ab
X
b
X—2m
3
2
(6m
)
4
b) n
)
)(3x
)(4x
(2xy
y
2
y
e) 3
b’‘
ci) 2
2n)
(5n
X
p)
p
3p
—’2
(—
h) 3
(12
)
5
X
521,
q
2
3p
45p
6
q
9
2
q
3
—14p
)
)(6x
3
y
2
(h
y
)(33
Z
4
(5y2(S
)
1) (5x2.v)N3xZ9P
FMPCIO
Unit 2: Polynomials
Lesson 2: Adding & Subtracting Monomials
In General: When adding and subtracting polynomials you can only add or subtract
like terms. This means that you can only add and subtract terms that
have the same variables raised to the same powers respectively.
Examøle 1: Simplify
a) (2x
—
2
c) (4x
—
7y)
+
(8x
7xy + 3)
3y)
—
2
(x
+
—
5yx
+
9)
—
2
(x
+
11)
b) (5x
+
6xy
11)
—
—
(2x
+
4)
—6x+2
2
7x
3x— 4
4x
—
2
d)Add:
Example 2: Simplify and state the degree of the monomial.
2
a) (4a
y
2
b) (x
—
—
)
2
3b
)
2
y
+
—
2
(2a
y
2
(4x
—
+
)
2
b
—
)
3x
y
2
2
(3a
—
+
2
(y
4)
—
=
y
2
2x
—
)
2y
x
2
=
HW:
2.2WorksheetAttached
2.2 Adding and Subtracting Monomials Worksheet
1. Simplify.
a) (6a÷9)÷(4a—5)
1,) (2x—7y)+(Sx—3y)
+4)(7+9m
(3m
)
c) 2
d) (+ll)—(2p+4)
—9x)—(6x
(17x
—
5x)
e) 2
1) (7k÷31)—(i2k—$1)
2. Simplify.
a) 2
—2m—7)
(3m
+
Sm+9)+(8m
3
b) (7a
—
2
3 + 6a
2 ÷ Sa) + (—3a
2a
—
2a)
—i5x—lly)
(x
+
Sx÷6y)—(4x
c) 2
d) (512_13t+17)+(9t3_712+31_26)_(l612+51_8)
+ 5a
3
)
2
3a
a 8)—(12 —7a
+(2a
—
2
(2÷8a
)
—
3
e) —
f) 2
6x—li)
—
(4x
—
+
Sx+9)—(8x
lx+3)—(x
4, Sirnplifi, and state the degree of the polynomiEL
y—5x
3x
33’X
+2y
—
(5x
y
x
—
+
)
—y)—(2xy
a) 2
2 ÷ 7q
q
3
p
2
b) (p
—
3g,) + (2q
—
q
2
6p
—
)
3
p
2
q
—
(3rn
-2n+7)
3rn+7)—(2n
c) 2
d) (2ab—2ac—2bc)÷(2ca÷2cb—2ba+3)
y
2
e) (xy + 3yx + 2x
—
)
2
3xy
—
(2xy
±
)
2
2yx
5. Mi
—6x±7
2
a) 3x
—2x÷9
2
7x
2
—
3
b)7z
÷
z—3
6z
_z±4
2
2z±4z
6. SWtract.
y—Jxy+2xy
5x
a) 2
v ÷ 4xy
2
3x
3
—
n
2
4mn÷3rnn
1) 6rn
3
n t2rntt 4mn
2
2rn
—
—
7. Sb plify, and state the degree of the polynomial:,
2y
2x
2)
—(4x
÷(y
(3x
)
—
)
÷
a) 2
m
n
m
÷(2n
—(n
—
(m
—
)
+
2
b) )
c) (4x
y
2
—
2yx) + (3yx
2
)— (3b
2
b —b
2
a
2
ci) (a
+
—
)
2
6xy
—
x
2
y ÷ 2y
2
(3x
b) —(b
2
2a
a
2
2 b
—
—
—
a)
2
b
FMPIO
Unit 2: Polynomials
Lesson 3: Multiplying Polynomials Part I
Example I: Simplify:
a) 2x(3xy—5)
y(4xy
3
-5x
b) 2
+
7x)
+
y
4
llx
c) 11(2x+ 3y)—7(3y— llx)
2
d) 2x(3x
—
5xy + y
)
2
—
x
5y
(
2
—
2y)
HW:
2.3 Worksheet Attached
2.3 Multiplying A Monomial and a Polynomial Worksheet
1. Smiify.
2 + 10)
a) 4(5x
b) 72a
—
c) -(3k
5)
—
2k)
+i?fl_
2
_9(...5ffi
)
e) 3
5p+
2
3(8p
)
1) 7
b) $(2a—3)÷lJa
c) 5(y+2)—7y
d) 4(lm—5)—13
2
e) —6(3p+2p)÷5p
f) 7(5x—3y)—$3x
a) 12 5x—4)
b) 3a(—7a+2)
c) 6p(2p—q)
e) 3
7rn
(
3nm +6)
—8x
(
5x ÷ ly)
f) 2
—.3b+9)
2
4) 12(2b
2. Simplify.
a) 3(x+4)+7
d) —15n
(6
2
—
9n)
4. Simplil.
a) 3(x +2) + 2(x —6)
c) 3(2a÷Wb—2c)—6(a—2b+Sc)
b) 2(x +9) 3(x + 7)
d) 3(2m—4n+3)—5(—2rn+5n—I)
5, Simplii’.
a) 3x
(x + y) + 2x
2
(3x + 5y)
3
b) 3a
(2a
3
—
—
Sb) — 4a
(2a + 3b
3
c) 2
5p
(
4p q) —8p
(2p —7q)
2
4) 6a
(—3a ÷ 7b —4) 8a
3
(2a 3b +7)
3
6. In a hockey league, each team plays afi other teams 4 iimes. Ifthere are n teams,
wiite a formula for the total number ofgames to be played.
—
—
7. Simp1if.
a) 2
(ab—a
—2ab
+
b)
—
y
b)3x
+
2
xy—y)
(xy
c) 2
n(3mn÷nm
—Sm
—
)
n
4) 5x(x—y)—231(x+y —1)+y
2
e) 2b(b—bc)—2c(b —c)÷(76c—4c
)
2
—
7x(x
)
÷
)
—2xy—2y(x
y
y
) 2
9. Simplity, and state the degree ofthe polynomiaL
a) 3aTh —?a(a —4)+(5ba
— 13a)
2
I,) 6rn(2mn
—
n)
c) 4xy(x + 2y)
—
—
n
2
3(m
—
19mn) + 6n(—Smn
—
)
2
7m
2 3y(3x
$x(2xy 3y
2 + 7xy
—
—
4) 2
—
5s(3E
+
+
s—6)+9(s
7s+2)+2s(5s
6s—4)
e) 4x(5x
2
1) Sxy—3y(2x —y)÷7x(Sy 2
—2y
3
—x)+9(x
)
10. Simpliê,
a) (1 la + 3X9a)
2 + 2a 3X—4a)
4) (5a
—
b) (3x SX—2x)
e) I 2n(3n —7)
—
c) $ 5x(16x + 12)
1’) 5x(2x
2 6x + 3)
—
FMPIO
Unit 2: Polynomials
Lesson 4: Multiplying Polynomials II
Example 1: Simplify:
a) (x+3)(x—2)
b) (2a—4)(a+2)
2
c) (x+3)
2
d) (2x—4)
—7x+12)
2
e) (x—3)(4x
f) (x —7) (x
+
8) (3x —4)
Example 2: Express the area of the shaded region below as a polynomial.
3x+2y
-*E- 2x—y
+y
3x-y
HW:
2.4 Worksheet Attached
2.4 Multiplying Polynomials Worksheet
1. Find the product.
a) (4x÷ lX2x+3)
d) (3x 7)(3x + 7)
(2c—3)4c—5)
f) 8x 2X3x 7)
b) (5in—2)6m+ 1)
e) 4r + 5X4t 8)
C)
—
—
—
2. Square the binomial.
b)(x+y)
a) (g+h)
f) (k—I)
e) (p...q)
c)(i+n)
g) (s•..f)
d)(b+c)
Ii) (u—v)
3. Find the product.
2
A +y)
3
a) (
4rn—7)
e) 2
(3a÷2)
c) 2
a) (5t+iO)2
2)
d) (Sx—9
Ii) (_5_6b)Z
2
b) (x+5y)
(—4y—1l)
f) 2
4. Find the product.
a) (2÷y)(3x+y)
d) (+3j)(+4i)
5. Find the product.
2
a) (3a + 5b)
(8x—3y)
e) 2
c) (2x—y)(3x+4y)
f) (6x—2yX3x—7y)
b) (3a+bX5a—b)
e) (Sin—2nXlm—n)
2
b) (2m 7n)
(—3s—5t)
f) 2
—
6. Find the product.
a) (x—y)(x+y)
d) 6x 2yX6x + 2y
—
—
b) (2m÷5n)(2rn—5n)
e) ç7p + 2q)7p 2q)
C)
fl
—
—
2
4) (4p 3q)
+7z
(5y
)
ii) 2
c) (6s 2i)3
(—5f÷4g)
g) 2
(9a—4bX9a+4b)
—12m + 9X—12m —9)
7. Simplify.
a) (x_3)x+2)+(2X—5XX I)
(2m÷3)(3rn÷5)+(rn—7X6m—
)
c) 3
+ 7b)
e) (2a 5b)(3a + b) (6a bX4a
b) (2x+4(3x—2)+(5X —2X3x—-4)
d) (c+2dXc+d)—(C—2d)(C—d)
f) (3x 2y)(5x y) (2x + 5’X3x y)
8. Simplify.
)
2+(x+
(
_S
(2x_3)
)
2
X
a) 3
2
2y)
(x
c) (2x y)(2x + y)
2
2 (5x + 2y)
e) (5x 2y)
)
S1)
2t
1)(4S—
—
-(s—3
2
(s
b) 2
d) (7in 2nX3in + 5n) (4rn I in)
2 ÷ 2)2
2 + 3Xx
2 5) (3x
f) (2x
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
9. Find the product.
÷ 3 ....4)
2
(3x 1X2x
a
—
5X3
8a
(2a—
+
2
)
c) 9
a)
—
10. Find the product.
a) (3x+4)(x—5X2X+8)
c) (3x ÷22x— 5)4x —3)
t2a 7)
2
e) 5a 3
—
—
m 12)
.3X4+
7m
(m—
—
b) 2
+
)(5p
6p
d)(3p+2
—
)
2
b) (4p—3X2p—7X5P—6)
d) (2x —5)(3x
5m
—
+4)2
—
11 Find the perimeter and area of each shaded region.
*
a)
r
1,)
F..
x-2yr’)1_jS+y
xty
F
Zx
k+y
x,
y
2
I
c)
I
.
.
1
H
r
—
1
x-y
12. Simplify,
a) 3(m—5)(m+2)
c) 5(2p—73p--4)
e) 3a(2a SbX3a + b)
b) 2(x÷yX3x—5y)
ci) 4x(x—ly)(2x—3y)
0 3x(2x +
—
13. Simplify.
a) 2(3x+lXx+5)÷4(2x+3XJx+5)
b) 5(2m 4X3m + 2)— 2(4m lX3m —4)
c) 3(2x + SyX7x $y) + 5(x ÷ 3y)(2x ÷ 4y)
d) 2(xy— 1)(xy÷ t)—5(xy—2)(xy+ 3)
e) 3(2a 3bXa + 2b) 2(3a 3)2
)
2)
y+5
y—
3(x
4(x
+
1) 2
—
—
—
—
—
—
14. Simplify.
a) (2x—$X3x+6)+(3x—2X4x÷9)—(Sx—3)(2x—7)
s
5(3
6)
(s+
—2)—
9)—
s÷5)(2s
7s—
2(3
+
b) 2
3x_7)
2xZ+x
)÷(x_4X3
Sx+4X
3
_
c) (2x+ 2
d) (3m+4Xm—4n—i)—(5m—2)(3m—6n—8)
) + 2X4y 5)2
(3y
+2
—
e) (
y 5)(3y 2
4
1)
5x
3x—
2X
f) 2
x+6X
x—
x÷6)—(2
4(3
÷
—
—
C-’
FMPCIO
Unit 2: Polynomials
Lesson 5: Factoring a GCF
2 (x + y)
2x
+
this process is called
=
this process is called
=
Where the
Example 1: Use algebra tiles to factor 2x
+
outside is the
8.
Step I
Model the polynomial 2x
Step 2
Arrange the tiles into a rectangle.
The total area of the tiles represents
the product.
+
—
8.
Example 2: Use Algebra Tiles to factor the following (sketch a diagram of solution).
a) 3x
+
9
b) 4x+1O
c) x
2 + 5x
2 + 6x
d) 3x
GCF for a Polynomial
GCF of the coefficients and the lowest power of any variable in every term.
Example 2: Find the GCF of the following:
a) 12, 18
y
3
, 6x
3
y
2
b) 3x
, I Ox, 35x
3
c) I 5x
Example 3: Factor
a) 12x+18
+
x=
5x
2
+
3
3x
b) —
60=
24m
2
+
4
36m
c) —
d) 5x(a—b)+7(a—b)=
e) 2m(2x+ 1)—5(2x+ 1)=
f) 6a(b—a)+4b(b—a)—7(b—a)=
HW: Pg.155#4,7-12,14,16
FMPCIO
Unit 2: Polynomials
Lesson 6: Factoring Trinomials Part I
Example 1: Factor the following using algebra tiles.
a) x
2 + 5x + 6
2 + Sx +4
b) x
2
c) x
—
5x + 6
+x—6
2
d) x
e) x
2
—
3x
—
10
Example 2: Factor Completely.
+12a+2O=
2
a) a
—18m+32=
2
b) m
+3n—40
2
c) n
—8y—20
2
d) k
+16x+14
2
e) 2x
27m
18m
+
f) 3m
-2
3
+l3xy+12y
x
g) 2
21
10a
+
h) a
-’- 2
4
)
+6(x+y)+8
2
(x+y)
j)
—4(x+2y)—12
2
(x-i-2y)
HW: Pg. 166#4,6,11,14,17,19-21
FMPIO
Unit 2: Polynomials
Lesson 7: Factoring Trinomials Part 2
Example 1: Factor the following.
2
a) 2x
+ 7x +6
+2x—3
2
b) 8x
+9x+4
2
c) 2x
+lOx+3
2
d) 8x
1. Multiply the outside coefficients
together.
2. Find two numbers that multiply to this
product and add to the middle
coefficient.
3. Set up brackets with the first
coefficient in the front position of
both sets and the two numbers that
you found in the last position.
4. Divide any common factors out of
each set of brackets.
5. Check by expanding
13x—1O
3x
—
e) 2
Example 2: Factor Completely.
26a
20a
2
+
3
6a
a) —
—l7xy—15y
4x
y
b) 2
+25xy—4y
21x
c) 2
2
d) 4x
—
2lxy
—
2
1 8y
=
HW:
Pg. 177 #12,13, 15, 18,19
FMPIO
Unit 2 Polynomials
Lesson 8: Factoring a Difference of Squares
Expand the following:
(x+ 3)(x—3)
(x
+
2) (x —2)
=
(3x—5)(3x+ 5)=
What’s the pattern???????????????????????????????????????????
To factor we will use this pattern and work backwards
Example 1: Factor the foNowing
—16
2
b)81m
—9
2
a)x
2
c) 1 6m
3
e) 1 2x
—
—
1 21n
2
27x
d) 6x
2
—
f) 63x
y
2
150
—
g) (5m
h) x
4
i) Y
4
—
—
—
2)2 —25
2 + 36
13x
2
5y
—
36
HW: Pg.194#6, 7, 10-13, 20, 21
2.1 Multiplying & Dividing Monomials Worksheet Key
La)35x’ b)l2tFc)—54m’ d)10n
3 e)56 O—28b
4
g$t’ h)72p’ 2.a)1$xy b)32pqc)35mn d)14a6
e)—45A? fl-44cd 2)63213 bc’ 3s)-Sab’
7 c)2:ia’b d)24jPq e)24x
n
5
b)-12rn
y’ f)—6a
6
b
4
4.a)4x’ b)9rn c)—9a’ 4)46 e)4cP f)4n
2 g)86
4
5a)2r432
4
2
b)10p
b)—3x’ c4p’q d)2a’b
e)—$m’ f)6x5t’ 6.a)54x” b)—12a’°b c$Ss’°s’
3 e)-24mn’ f)675xyt4 7a)&’y
d)—lOOp’q
3
4 e)—16rn’n’ fla’
b)—6? c4a263 d)2x
b”
3
2.2 Adding and Subtracting Monomials Worksheet Key
tsci tcc.1
.
K
ey.
‘J
÷l
1oYO1—o—
rnp
3
1.fl÷4bnÜE—
4
3m + 2
— 4x fl—5k + 11, 2.a11n$
)—3Z I7y
i+
b)4d+4d+3aC
—
3
0ic2 — Sr ÷ 5
— 18
— 189 — 1St — I e)Sd
30 000 1,1547 500; 3125 000; 5280 0001
3a)P= 15.Sn
4,a)1(le — 6x& third b)p’q’ 3p + Zq; fourth
2,9 + 2n;sccond d)3;zero
- 3m
e)—3xf + 2xy third LaWh’ Sx ÷ 16
b)9z’—2t+1 4s)2fl—7Xy+52Y’
- 2; second
6,wz + 7aut 7.s)-3x b)4mW
; fourth
2
6xy
b)3m’; second c)-Sfl + 7zy 8.Answers
fourth
;
1
2b
2ab’
2a’b
d)2d9
may vary.
—
—
—
—
—
-
-
-
-
-
2.3 Multiplying A Monomial and a Polynomial Worksheet Key
Eerdses 3-4, page 85
2
i.a)2C1t+40 b)14a—35 c)lók-24k
63m +27
2
e)45m
108
+
36b
2
d)24b
15p €21 2a)3x + 19 b)24 5a eflO 2y
fl24
12p fl4x 21y
4)28m —33 e)—13
1 — 6pq
2 e)12p
2 48x b)6a 2hz
3.a)60x
3
3 f)—5’—40x
n +42rn
4
d)135n’—90n e)21m
4s)Sx —6 b)-x —3 c)42b 36c dN6rn 5Th ÷14
3 + Slp’q
—27a c)4p
4
b)—2a
6
y3
2
5.t)9x’ + 13x
—1)
6.2n(n
80a’
:d>..34a’ + 66a
6
3
2
y
3
3 + 3x
3 b)3x’y
2 24261_ 2ab
b
3
7.sfl2a
7 +2y
2 d$x
n
3
2
n’ Sm’,? 15m
2
c)Sm
2b She
--29+
3
e)2b
c
2
3 -lx/ fly Zx’y 2 8a)66 b)Z’
f)7x
6 72 + iSa; third b)21 mit’ 33m’n; thir:
3
9s)8a
rd
+21s
38s;thi
d)43s’
—
y; third 2
2
c)lxy’- iSx
y 53vtbird
2
3 14x
e)5x
f)SSxy 15,2 + 2x’; second
2 c)40x
2 + 30x
iO.a)99ê + 27a b)1Ox :6x
d)—20a’ 8a’ + 12a c)3.6n’ 8Air
+ 15x ils)29p— i8q+29
2
fllOx’—30x
b)—29k—41—75 c)3x’+Sxy—9x—29— i4y
y—2x’
2
d)Sa”- lOOa e)45x’—8x
n—m’—94mn 12.a)a+5b b)lOa÷4b
2
f)30m
c)a’÷b’ d)a’—a’b—ah’ —6’ 134 =2icrr÷n)
— IGn d)3&+ 14n
2
2 b$& c)8xr
14s)911 cm
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
-
-
-
—
—
—
—
—
—
—
—
2.4 Multiplying Polynomials Worksheet Key
Key:
n —2
7rm
b)30
—
)8x 14x+3 2
2
I.a
+
4
e)16t 12t—40
—
49 2
d)9x
—
c 15 2
22c+
c)8
—
2
2 ÷ 2xy +
2 b)x
2.a)g + 2gh + h
f)24x 62x + 14 2
2
q
c
+n
÷
÷
b
p
q
n
c
m
p
)
b
)
m
)
2
2
2
c
+
d
+
e
—
1
2
2 2uv +
2 2sf + 12 h)u
2 2k! + j2 g)s
f)k
5y
2 + lOxy + 2
2 b)x
3a)9x + 6xy + y
2
d)25x 90x + 81
c)9a + 1 2a + 4 2
2
8 121
1
+
+
f)
9
6
y
9 8
rn
6+4
56m
e)1
—
2
2
g)25r 1001 + 100 h)25 ÷ 60b + 36b
2
y
b
x
a
÷
6
5
—
)
y
b
a
1
x
a
.
)
.
S
42
+
b
+
2
2
y+ 12y
)x
7x
d
÷
x 2
y-4y
)6
5x
c
+
2
2
f)18x 48xy + 14y
2 2
e)35m I9mn + 2n
2
2
b)4m 28mn + 49n
2 2
5.a)9a + 3Oab +4225b
2
2
d)16p 24pq + 9q
2
c)36s 24sf +
2
2512
f)9s + 30s1 +
2 2
e)64x 48xy + 9y
2
2
70y + 49z
1
z
h)25y + 2
2 4
g)25f —4Ofg + 6g
2
6.a)x —y b)4m —25n c)81a —16b
f)144m 81
2 2
2 2
e)49p 4q
d)36x 4y
2
c)12m 26m + 36
b)21x 18x 2
7a)3x 8x I 2
2
y
x
xy+7
269
f)
—
5lab+2b 2
18a 2
e)—•
d)6cd 2
2
b)—3s l3st 1Ir
÷
8a)7x lix— 1 2
—
2
d)5m ÷ lllmn 131,12 e)—4Oxy
2 2
c)3x + 4xy 5y
2
2 I 5x + 4
— 19 9.a )6x 4 7x
2
f)—7x 1 9x
4
b)4m —19m +33m-r36c)6a +a —58a÷45
2 128x 160
9)aJEx + 2x
2 6p + 4 3
d)15p 8p
3
O
x
x
3
÷2
4
6
x
2
-7
)
—
3
6 2
p
p
12
8
0
—
1
4
2
9p
)
30
b
—
3
÷
2
—235a + 228a —63
e)50a 2
÷3x 88x —80 3
—
d)18x 2
3
8 11a)iOx+2y;
m
0rn
—
5
5
2
m
1
0
1
6
f)
—
3
+
2
2
2 + l4xy + 2y
2 + 2xy 4$ b)14x + 6y; lOx
4x
m
)330
.,a—
129m
—
x 2
20
2
x+
8;1
138
+
c)16x+ 2
c)30p 145p + 140
2 2
b)6x 4xy 10y
2
15ab
39a 2
b
e)18a 2
84xy 3
68x + 2
y
d)8x 2
3
48xy
48x + 2
y
f)12x + 2
3
8
rn
2m--4
b)6
—
13a)30x 108x+70 2
+
2
d)—3x 5xy + 28
y
c)52x + IOlxy 609 2
2
f)7x l4xy+91
+
y
l5ab—20b 2
e)—12a 2
÷
2
b}-4s 39s 11
14..a)8x + 57x 69 2
2
2 26x + 40
c)8x 6x
3
d)—12m ISmn ÷47m —28n—20
÷
2
2 + 98x 54
f)42x 70x
2 39y 70 3
e)— 1 2$ + 91 v
b)3m + 16m + 28
15a)m + tim + 26 2
2
2
16a)4a + 4ab + 4b
2 + 6m 5 2
c)rn + 5m
3
2
c)8a ÷ 6ab ÷ 2b
2 2
b)—5a + l2ab 3b
2
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
-
—
—
--
—
—
—
—
—
—
•
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
ID:A
Polynomials Review
Answer Section
MULTIPLE CHOICE
SHORT ANSWER
1. B
2. A
3. B
4. B
5. A
6. C
7. C
8.A
9.
10.
11.
30. (s—32)(s---l)
31. 7(2z—5)(z—l)
27
+
3
+
2
6Z
102
0
+1
z
z
32. 2
33. (7s+8t)(7S—8t)
)
9
8(z—5)(z—
35. =4(7x—2y)(7x+2Y)
D
C
36.
The outer rectangle area
=
6
y
9
(
y
)(
)
The inner rectangle has area
13. B
=
2
54y
(5y)(3y)
=
2
15y
b) The total area of the shaded region is the difference in the areas of the rectangles:
14. C
54
—
=
2
39y
15y
y
15.
17
18.
19.
20.
21.
22.
a)
B
A
D
B
B
24. D
25.A
26. B
27. B
28. B
29. B
37.
2
a)6
—
llm+3
m a) 4y
—25 c) 2
2
+l3xy+6y d) 2
6x
—
4
2a
—
10
a e) 8x
—16x+4
2
2 —6p+4 g) —22m h) —6m
f) 2p
22
—5mn+9m+4n+2n i) lOx
2 —llx+3 j) 9x
2 —49
k) 32x
+16x+2 1) 2p
2
—6p+4 m) 6x+20y
2
38.
a) (x+4)(x+3) b) (m—5)(m+2) c) 2
(2k÷l) d) (2p—3)
2 e) 2
(6x—
)
)
(6x+5y
f) 2
(x—2)(x+2)(x +4) g) 5(2x—3)
2 h) (—x—1)(3x+1) i) (m—3n)(3m+4n) j) (x+2)(5x—1)
k) (x—S)(x—4) 1) (2x—3)(3x+4) m) (x+3)
2 n) (m—9)(m+2) o) (6k+1)
2
2 —6p+8) q) 2
p) 2(p
(3x—2y)(3x+2y)(9x +2y
) r) (x-13)(x+13) s) (x—2)(4x—7) t) —4(y—l)(2y
2
u) not factorable (prime) v) (x + 3)(x —4— Sy) w) (x y 7)(x y + 2)
—
4
—
—
ID:A
Date:
Class:
Name:
U2
Polynomials Review
Multiple Choice
Identjfr the choice that best completes the statement or answers the question.
1. Simplify the expression
, then factor.
6
2 —24y—2
2 +8y—6—9y
y
a.
2y—4)
—8(y
—
2
b.
2y+4)
—8(y
+
2
c.
4y+8)
—4(2y
+
2
d.
4y+1)
—4(2y
+
2
2. Expand and simplify: (p
a.
—4p—21
2
p
b.
—lOp—21
2
p
c.
lOp—21
p
+
2
d.
4p—21
p
+
2
+
7. Expand and simplify: (5m
a.
b.
c.
d.
—
2
3n)
—9n
2
25m
—lSmn+9n
25m
2
2
2 —3Omn+9n
25m
+9n
25m
2
8. Each shape is a rectangle. Write a polynomial, in
simplified form, to represent the area of the shaded
region.
(p —7)
)3
3. Expand and simplify: (6p + 3)(5p —6)
a.
21p—18
30p
+
2
b.
21p—18
30p
—
2
c.
51p—18
30p
+
2
d.
51p—18
30p
—
2
a.
b.
c.
d.
4. Expand and simplify:
(3c + 2)(2c —7) + 3(—2c + 1)(7c —5)
a.
b.
c.
d.
8c—29
—36c
+
2
+34c—29
2
—36c
—8c—19
2
—36c
—8c—29
2
—36c
31x+66
5x
+
2
+37x+30
2
5x
+31x+30
2
5x
+37x+66
2
5x
9. Which polynomial, written in simplified form,
represents the area of this rectangle?
8x
2
2 + 5x 6)(5x
5. Expand and simplify: (2x
2 +27x—18
3 —34x
4 +21x
a. lOx
34x
3x+18
+21x
4
10x
—
3
—
b. 2
2 +27x+ 18
3 —24x
4 +21x
c. lOx
3
—29x
2 +27x— 18
—34x
4
d. lOx
—
2
6. Expand and simplify: (8h + 3)(7h
2 —20h+3
3 —53h
a. 56h
12h+3
÷11h
3
56h
—
b. 2
4h+3
11h
—
3
56h
—
c. 2
32h
8h+3
—
3
56h
+
d. 2
—
—
—
4y
x + 5y
2x + 3)
4h + 1)
1
a.
—36xy—20y
2
8x
b.
+22xy—20y
2
8x
c.
+72xy—40y
2
16x
d.
+36xy—20y
2
8x
ID:A
Name:
)=k
Dk+135
(k—D)(k—5
—
17. Complete. 2
2)=k
5k+135
(k—27)(k—2
—
a. 2
.
2
10. Factor the binomial 44a + 99a
a. a(44+99a)
b.
c.
d.
11(4a+9a
)
2
lla(4+9a)
22a(2+9a)
c.
)=k
32k+135
2
(k—27)(k—5
—
2)=k
5k÷135
(k—27)(k—3
—
2
d.
)=k
22k+135
2
(k-.27)(k—5
—
b.
2 + 99b + 77.
11. Factor the trinomial —33b
—9b+7)
2
a. —11(3b
b.
3b—7)
—33(b
—
2
c.
9b--7)
—11(3b
—
2
d.
27b÷7)
33(—b
+
2
2
2d
3 d 40c
12. Factor the trinomial —24c
)
2
2 5cd—4d
a. —8cd(3c
—
2 14n —105
18. Factor: 7n
a. 7(n+3)(n—5)
b. 7(n+15)(n—1)
c. 7(n—15)(n+1)
d. 7(n—3)(n+5)
—
—
19. Which multiplication sentence does this set of
algebra tiles represent?
.
3
32cd
—
b.
)
2
2 +5cd+4d
8cd(3c
c.
)
2
2 +5cd+4d
8cd(—3c
d.
)
2
2 +5cd+4d
—8cd(3c
13. Identify the greatest common factor of the terms in
42
34
t
15s 23
the trmomial 6s t + 1 2s t
22
a. 6st
t
2
b. 3s
3
t
2
c. 3s
2
t
3
d. 3s
—
+9t—36
2
14. Factor: t
a. (t—2)(t+ 18)
b. (t+2)Q—18)
c. (t+12)(t—3)
d. (t—12)Q÷3)
15. Factor: —84+8z+z
a. (42+z)(—2+z)
b. (—6+z)(14+z)
c. (—42+z)(2+z)
d. (6+z)(—14+z)
a.
(2x—2)(2x+2)
b.
(2+2)(2÷2)
c.
d.
(2x-f-2)(2x+2)
2 ÷ 58x + 16
20. Factor: 25x
a. (25x+4)(x+4)
b. (25x+8)(x+2)
c. (5x+4)(5x+4)
d. (5x+8)(5x+2)
2
2 ll6y+6O
21. Factor: 48y
a. (l6y—l2)(3y—5)
b. 4(4y—3)(3y—S)
c.
d. 4(4.y+3)(3y+5)
—
2 28d + 240
16. Factor: —4d
a. —4(d + 3)(d 20)
b. —4(d+5)(d—12)
c. —4(d—3)(d+20)
d. —4(d—5)(d+12)
—
—
2
ID:A
Name:
27. Determine the area of the shaded region in factored
form.
2 + 324w —42
22. Factor: 96w
a. 6(8w+1)(2w—7)
b. 6(8w+7)(2w—1)
c. 6(8w—7)(2w+1)
d. 6(8w—1)(2w+7)
+llOa+25
2
23. Factor: 121a
a. (lla+5)(lla—5)
b. (121a+5)(a+5)
c.
d.
2
(ha—S)
2
(lla+5)
a.
b.
c.
d.
—81q
16p
24. Factor: 2
2
a. (4p—9q)
b.
c.
d.
2
(4p+9q)
(16p—9q)(p—9q)
(4p+9q)(4p—9q)
4(x+12)
(3x+2)(x+12)
(3x+12)(x+2)
(3x—2)(x—12)
2 +k+1 be
28. For which values of k can 6x
factored?
768z
4 2
25. Factor: 3z
(z+ 16)(z— 16)
2
a. 3z
(z+ 16)2
2
b. 3z
—
a.
,
7 only
z+48)(z—16)
(z
2
b.
±5, ±7 only
c.
d.
(z—16)
2
3z
c.
-5, -7 only
d.
all integers between -7 and 5, inclusive
26. Factor Completely: 162
)(18—w
(9—w
)
a. 2
—
4
2w
b.
(3+w)(3—w)
2(9+w
)
2
c.
)
2
2(9—w
d.
)
2
2(9+w
29. Which of the following is a factor of the
2 + 2x —8?
polynomial 3x
a.
b.
c.
d.
3x+2
3x-4
3x+4
x-2
Written
30. Factor:
2
S
34. Factor. Check by expanding.
2 112z+ 360
8z
—33s+32
—
49z+35
14z
—
31. Factor: 2
2
32. Expand and simplify: (9z
2
33. Factor: 49s
—
35. Factor.
—
2
196x
2z + 10)(3z + 12)
2
64t
3
—
2
16y
ID:A
Name:
36. The diagram below shows one rectangle inside
another.
a) Determine the area of each rectangle.
b) Determine the area of the shaded region.
38.
Factor Completely
+7x+12
2
a) x
-3m-1O
2
6) m
+4k+1
2
c) 4k
—12p+9
2
d) 4p
e)
—25y
2
36x
4
f)
x”—16
20x
+
2
60x+45
2
6) x2_(2x+1)
g)
9
37. Simplify the following:
j)
k)
a)
(2m—3)(3m—1)
6)
(2y—5)(2y+5)
c)
(2x+3y)(3x+2y)
(a2
+2)(2a2
m)
n)
o)
5)
e)
2
4(2x—1)
I)
2(p —1) (p —2)
g)
5(—2m--3n)—3(4m—5n)
6) —3m (2m + n —3)— 2n (m
9
—
p)
q)
—169
2
r) x
—15x+14
2
s) 4x
n —2)
t)
(3x+7)(3x—7)
2(4x+1)
1)
2(p—1)(p—2)
2
y2_(3y_2)
—l8mn—8n
9m
u) 2
v) x(x+3)—4(x+3)+5y(x+3)
(5x—3)(2x—1)
)
2
3m —5mn—12n
2
+9x—2
2
5x
x(x—5)—4(x—5)
—x—12
2
6x
+6x+9
2
x
—7x—18
2
m
+12k+1
2
36k
—12p+16
2
2p
—16y
81x
4
—5(x—y)—14
2
w) (x—y)
m) —3(2x—4y)+4(3x+2y)
4
ID:A
Polynomials Review
Answer Section
MULTIPLE CHOICE
1. B
2.A
3. B
4. B
5.A
6. C
7. C
8.A
9.D
10. C
11. C
12.D
13. B
14. C
15. B
16.D
17. B
18.A
19.D
20. B
21. B
22.D
23.D
24.D
25.A
26. B
27. B
28. B
29. B
SHORT ANSWER
30. (s—32)(s—1)
31, 7(2z—5)(z—1)
2 +6z+ 120
3 + 102z
32. 27z
33. (7s+8t)(7s—8t)
34. 8(z—5)(z—9)
35. =4(7x—2y)(7x+2y)
ID:A
36.
a)
The outer rectangle area
=
(9y)(6y)
The inner rectangle has area
=
=
2
54y
(5y)(3y)
=
2
15y
b) The total area of the shaded region is the difference in the areas of the rectangles:
2
—15y =39y
54y
2
37.
2 —16x+4
4 —a
2 —10 e) 8x
2 +l3xy+6y
2 d) 2a
2 —25 c) 6x
a) 6m
2 —llm+3 a) 4y
—49
2
2 —llx+3 j) 9x
—5mn+9m+4n+2n i) lOx
—6m
f) 2p
—6p+4 g) —22m h) 2
2
—6p+4 m) 6x+20y
2
+16x+2 1) 2p
2
k) 32x
)(6x+5y
(6x—5y
)
2 e) 2
(2k+l) d) (2p—3)
a) (x+4)(x+3) b) (m—5)(m+2) c) 2
2 h) (—x—1)(3x+l) i) (m—3n)(3m+4n) j) (x+2)(5x—l)
2 +4) g) 5(2x—3)
f) (x—2)(x+2)(x
38.
(6k+l)
2 n) (m—9)(m+2) o) 2
k) (x—5)(x—4) 1) (2x—3)(3x+4) m) (x+3)
y—l)
2
) r) (x-13)(x+13) s) (x—2)(4x—7) t) —4(y—l)(
2
2 +2y
2 —6p+
) q) (3x—2y)(3x+2y)(9x
8
p) 2(p
u) not factorable (prime) v) (x+3)(x—4—5y) w) (x—y—7)(x--y+2)
2