Download Chapter 3 - Sardis Secondary

Chapter 3
Factoring
1. Write the prime factorization of 630.
630
2
315
5
63
3
21
3
= 2 x 32 x 5 x 7
7
2. Determine the greatest common factor of 56 and 88.
56
1, 2, 4, 7, 8, 14, 28, 56
88
1, 2, 4, 8, 11, 22, 44, 88
2. Determine the greatest common factor of 56 and 88.
88
56
2
28
2
14
2
44
2
2
7
GCF = 2x2x2 = 8
22
2
11
3. Determine the least common multiple of 10 and 22.
10
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120,
22
22, 44, 66, 88, 110, 132, 154, 176, 198, 220, 240
3. Determine the least common multiple of 10 and 22.
10
2
22
5
2
=2x5
11
= 2 x 11
Circle prime factors so the highest
power of each prime is selected, then
multiply those to find LCM
LCM = 2x5x11
LCM = 110
4. Determine the edge length of this cube.
V    w h
V = (x)(x)(x)
Volume
= 91 125 cm 3
91125 = x3
3
91125  x
x = 45 cm
5. Factor the binomial
44a  99a
2
= 11a(4 + 9a)
6. Factor the trinomial
 24c d  40c d  32cd
3
2
2
= -8cd(3c2 + 5cd + 4d2)
3
7. Expand and simplify: 5m  3n 2
5m  3n(5m  3n) = 25m2 – 15mn – 15mn + 9n2
= 25m2 – 30mn + 9n2
8. Expand and simplify:
8h  3(7h2  4h  1)
= 56h3 – 32h2 + 8h +21h2 – 12h +3
= 56h3 – 11h2 – 4h +3
9. Expand and simplify:
2x
2

 5x  6 (5x  2 x  3)
2
= 10x4 – 4x3 + 6x2 + 25x3 – 10x2 + 15x – 30x2 + 12x – 18
= 10x4 +21x3 – 34x2 + 27x – 18
10. Expand and simplify: 6 x  y 3x  8 y   2 x  3 y 
2
6x  y3x  8 y  2x  3y 2x  3y
= (18x2 + 48xy – 3xy – 8y2) – (4x2 – 6xy – 6xy +9y2)
= (18x2 + 45xy – 8y2) – (4x2 – 12xy +9y2)
= 14x2 + 57xy – 17y2
11. Factor the following:
Step 1
-12
a)
x  4 x  12
2
= (x + 6)(x – 2)
2
3
Is there a common
factor? No
Multiply (+1)(-12)
Look for numbers:
___ x ___ -12
___ + ___ +4
+6 & -2
4
Split into Brackets.
First coefficient is
always the same as
original expression.
5
Divide by the GCF in
each bracket
11. Factor the following:
+36
b)
9c 2  12c  4
= (9c – 6)(9c – 6 )
3
3
= (3c– 2)(3c– 2)
Step 1
2
3
Is there a common
factor? No
Multiply (+9)(+4)
Look for numbers:
___ x ___ +36
___ + ___ -12
-6 & -6
4
Split into Brackets.
First coefficient is
always the same as
original expression.
5
Divide by the GCF in
each bracket
= (3c– 2)2
11. Factor the following:
Step 1
c) 24b 2  50b  14
-84
= 2(
12b2
+ 25b – 7)
= 2(12b + 28)(12b – 3 )
4
3
2
3
Is there a common
factor? Yes
Multiply (+12)(-7)
Look for numbers:
___ x ___ -84
___ + ___ +25
+28 & -3
4
Split into Brackets.
First coefficient is
always the same as
original expression.
5
Divide by the GCF in
each bracket
=2(3b + 7)(4b – 1)
11. Factor the following:
Difference of Squares
d)
49 s  64t
2
Step 1
2
2
= (7s + 8t)(7s – 8t)
3
Is there a common
factor? No
Take the square root
of both terms and
separate into two sets
of brackets.
One Positive and
One Negative
11. Factor the following:
-40
e) 8c 2  18cd  5d 2
= (8c + 20d)(8c – 2d)
2
4
=(2c + 5d)(4c – d)
Step 1
2
3
Is there a common
factor? No
Multiply (+8)(-5)
Look for numbers:
___ x ___ -40
___ + ___ +18
+20 & -2
4
Split into Brackets.
First coefficient is
always the same as
original expression.
5
Divide by the GCF in
each bracket
11. Factor the following:
Difference of Squares
f)
Step 1
2
3 z 4  768 z 2
= 3z2( z2 – 256)
= 3z2(z + 16)(z – 16)
3
Is there a common
factor? Yes
Take the square root
of both terms and
separate into two sets
of brackets.
One Positive and
One Negative
12. Calculate the area of the shaded region
Large Rectangle:
2x  33x 1
= 6x2 – 2x + 9x – 3
= 6x2 + 7x – 3
Small Rectangle:
AShaded = (6x2 + 7x – 3) – (2x2 – x)
x2 x  1
= 2x2 – x
= 4x2 + 8x – 3
Algebra Tiles
-
2
+x
+x
2
x
+x
+1
-x
-x
-1
13. Draw the following factors using algebra tiles.
There is a legend on your formula sheet:
+6
a)
This is the number
of small tiles
x  5x  6
2
(x + 2)
This is the number
of big tiles
Step 1: Factor
= (x + 3)(x + 2)
(x + 3)
Represents the
long skinny tiles
13. Draw the following factors using algebra tiles.
There is a legend on your formula sheet:
This is the number
of small tiles
+2
b)
x  3x  2
2
= (x – 2)(x – 1)
This is the number
of big tiles
(x – 2)
(x – 1)
Step 1: Factor
Represents the
long skinny tiles
13. Draw the following factors using algebra tiles.
There is a legend on your formula sheet:
-12
c)
This is the number
of small tiles
Step 1: Factor
2x  x  6
__ x __ = -12
__+__ = +1
2
= (2x + 4)(2x – 3)
This is the number
of big tiles
(x + 2)
(2x – 3)
2
1
= (x + 2)(2x – 3)
Represents the
long skinny tiles
13. Draw the following factors using algebra tiles.
There is a legend on your formula sheet:
-3
d)
This is the number
of small tiles
x  2x  3
__ x __ = -3
__+__ = -2
2
= (x + 1)(x – 3)
This is the number
of big tiles
(x – 3)
(x + 1)
Step 1: Factor
Represents the
long skinny tiles