Chapter 3

Factoring

1. Write the prime factorization of 630.

630

2 315

5 63

3 21

3 7

= 2 x 32 x 5 x 7

56

881, 56 2, 28, 4, 14, 7, 8,

1, 88 2, 44, 4, 22, 8, 11,

2. Determine the greatest common factor of 56 and 88.

56

2 28

2 14

2 7

GCF = 2x2x2

2. Determine the greatest common factor of 56 and 88.

88

2 44

2 22

2 11

= 8

10

2210, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110,

22, 44, 66, 88, 110, 132, 154, 176, 198, 220, 240

3. Determine the least common multiple of 10 and 22.

120,

3. Determine the least common multiple of 10 and 22.

LCM = 2x5x11

10

2 5

22

2 11

LCM = 110

Circle prime factors so the highest power of each prime is selected, then multiply those to

find LCM

= 2 x 5 = 2 x 11

4. Determine the edge length of this cube.

Volume3= 91 125 cm

V = (x)(x)(x)

hwV

91125 = x3

x3 91125

x = 45 cm

5. Factor the binomial 29944 aa

= 11a(4

6. Factor the trinomial 3223 324024 cddcdc

= -8cd(

3c2+ 5cd+ 4d2)

+ 9a)

7. Expand and simplify: 235 nm

= 25m2

– 15mn )35(35 nmnm – 15mn + 9n2

= 25m2

8. Expand and simplify:

= 56h3– 32h2

)147(38 2 hhh

+21h2

= 56h3

+ 8h – 12h +3

– 30mn + 9n2

– 11h2 – 4h +3

9. Expand and simplify:

= 10x4– 4x3

)325(652 22 xxxx

+ 6x2+ 25x3 – 10x2 + 15x– 30x2 – 18

– 34x2

+ 12x

= 10x4+21x3 + 27x

– 18

10. Expand and simplify:

= (18x2

+ 48xy

yxyxyxyx 3232836

– 3xy

– 8y2) 4x2 – 6xy +9y2)

= 14x2

– 6xy

= (18x2 + 45xy – 8y2)

– (4x2 – 12xy +9y2)

+ 57xy – 17y2

232836 yxyxyx

– (

11. Factor the following:

a) 1242 xx

= (x )(x )+ 6 – 2

No

-12

+6 & -2

Is there a common factor?Step 1

Multiply (+1)(-12)

2

Look for numbers: ___ x ___ -12 ___ + ___ +4

3

4 Split into Brackets. First coefficient is always the same as original expression.Divide by the GCF in each bracket

5

11. Factor the following:

b) 4129 2 cc

= (9c )(9c )– 6 – 6

No+36

-6 & -63 3

= (3c– 2)(3c– 2)

Is there a common factor?Step 1

Multiply (+9)(+4)

2

Look for numbers: ___ x ___ +36 ___ + ___ -12

3

4 Split into Brackets. First coefficient is always the same as original expression.Divide by the GCF in each bracket

5

= (3c– 2)2

11. Factor the following:

c)

= 2(12b )(12b )+ 28 – 3

-84

+28 & -3

4 3

=2(3b + 7)(4b – 1)

Is there a common factor?Step 1

Multiply (+12)(-7)

2

Look for numbers: ___ x ___ -84 ___ + ___ +25

3

4 Split into Brackets. First coefficient is always the same as original expression.Divide by the GCF in each bracket

5

145024 2 bb

Yes

= 2( 12b2 + 25b – 7)

11. Factor the following:

d)= (7s 8t)(7s 8t)+ –

Is there a common factor?Step 1

Take the square root of both terms and separate into two sets of brackets.

2

One Positive and One Negative

3

No

22 6449 ts

Difference of Squares

11. Factor the following:

e)

= (8c d)(8c d)+ 20 – 2

-40

+20 & -2

4 2

=(2c + 5d)(4c – d)

Is there a common factor?Step 1

Multiply (+8)(-5)2

Look for numbers: ___ x ___ -40 ___ + ___ +18

3

4 Split into Brackets. First coefficient is always the same as original expression.Divide by the GCF in each bracket

5

No22 5188 dcdc

11. Factor the following:

f)

Is there a common factor?Step 1

Take the square root of both terms and separate into two sets of brackets.

2

One Positive and One Negative

3

YesDifference of

Squares24 7683 zz

= 3z2( – 256)z2

= 3z2(z 16)(z 16)+ –

12. Calculate the area of the shaded regionLarge Rectangle:

Small Rectangle:

1332 xx

= 6x2– 2x + 9x

= 6x2 + 7x – 3

– 3

12 xx

= 2x2– x

AShaded = (6x2 + 7x – 3)

– (2x2 – x)

= 4x2+ 8x– 3

Algebra Tiles

+x2 -x2

+x+x-x

-x

+1

-1

13. Draw the following factors using algebra tiles. There is a legend on your formula sheet:

a)652 xx

Step 1: Factor

= (x + 3)(x + 2)

+6 This is the number of small tiles

This is the number of big tiles

Represents the long skinny tiles(x +

3)

(x +

2

)

13. Draw the following factors using algebra tiles. There is a legend on your formula sheet:

b)232 xx

Step 1: Factor

= (x – 2)(x – 1)

+2This is the number of small tiles

This is the number of big tiles

Represents the long skinny tiles

(x – 2)

(x –

1)

13. Draw the following factors using algebra tiles. There is a legend on your formula sheet:

c)62 2 xx

Step 1: Factor

= (x + 2)(2x – 3)

-12This is the number of small tiles

This is the number of big tiles

Represents the long skinny tiles

(2x – 3)

(x +

2

)

__ x __ = -12__+__ = +1

= (2x + 4)(2x – 3)2 1

13. Draw the following factors using algebra tiles. There is a legend on your formula sheet:

d)322 xx

Step 1: Factor

-3This is the number of small tiles

This is the number of big tiles

Represents the long skinny tiles

(x – 3)

(x +

1

)__ x __ = -3__+__ = -2

= (x + 1)(x – 3)