1

Unit 3 Factors & Products

General Outcome: • Develop algebraic reasoning and number sense.

Specific Outcomes:

3.1 Demonstrate an understanding of factors of whole number by determining the:

o prime factors

o greatest common factor

o least common multiple

o square root

o cube root

3.2 Demonstrate an understanding of the multiplication of polynomial expressions (limited to

monomials, binomials, and trinomials), concretely, pictorially, and symbolically.

3.3 Demonstrate an understanding of common factors and trinomial factoring, concretely,

pictorially, and symbolically.

Topics:

• Prime Factorization, Greatest (Outcome 3.1)

Common Factor, Lowest Common

Multiple

• Multiplying Polynomials (Outcome 3.2)

• Common Factors (Outcome 3.3)

• Factoring by Grouping (Outcome 3.3)

• Factoring by Decomposition (Outcome 3.3)

• Factoring Strategies (Outcome 3.3)

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Unit 3 Factors & Products

Prime Numbers:

Prime numbers are numbers that have exactly two divisors, 1

and itself. In other words, these are numbers that can only be

divided evenly by 1 and itself.

Ex) 7, 11, 19

**The number 1 is not considered prime as it only has one

number it can be divided by evenly.

Ex) List all of the prime numbers up to 20.

Composite Numbers:

Composite numbers are numbers that have more than two

divisors or are numbers that can be evenly divided by more than

two numbers.

Ex) 12, 6, 45

3

Prime Factorization:

The prime factorization of a natural number is the number

written as a product of its prime factors.

Ex) Determine the prime factorization of the following

numbers.

30 24

72 5000

144 90

4

Greatest Common Factor:

The greatest common factor of two or more numbers is the

largest number that each can be divided by.

Ex) Determine the greatest common factor for each set of

numbers.

a) 12 and 20

b) 138 and 198

c) 126 , 144, and 630

5

Least Common Multiple:

The least common multiple is the smallest number that can be

divided evenly by two or more given numbers.

Ex) Determine the least common multiple of each set of

numbers.

a) 14 and 21

b) 18, 20, and 30

c) 28, 42, and 63

6

Ex) What is the side length of the smallest square that could be

tiled with rectangles that measure 16 cm by 40 cm?

Assume the tiles cannot be cut.

Ex) What is the side length of the largest square that could be

used to tile a rectangle that measures 16 m by 40 m?

Assume the squares cannot be cut.

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Prime Factorization / Greatest Common Factor /

Lowest Common Multiple Assignment:

1) List the first 6 multiples of each number.

a) 6 b) 13 c) 22

d) 31 e) 45 e) 27

2) Express each number as a product of its prime factors.

a) 40 b) 75 c) 81

d) 120 e) 140 f) 192

g) 45 h) 160 i) 96

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3) Explain why the numbers 0 and 1 have no prime factors.

4) Determine the greatest common factor for each set of numbers.

a) 45, 84 b) 64, 120 c) 81, 216

d) 180, 224 e) 160, 672 f) 220, 860

g) 150, 275, 420 h) 120, 960, 1400

i) 126, 210, 546, 714 j) 220, 308, 484, 988

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5) Determine the lowest common multiple for each set of numbers.

a) 12, 14 b) 21, 45 c) 45, 60

d) 38, 42 e) 32, 45 f) 28, 52

g) 20, 36, 38 h) 15, 32, 44

i) 12, 18, 25, 30 j) 15, 20, 24, 27

6) When is the product of two numbers equal to their least common multiple?

10

7) Use greatest common factors or prime factorization to simplify the following

fractions.

a) 185

325 b)

340

380 c)

650

900

d) 840

1220 e)

1225

2750 f)

2145

1105

8) A developer wants to subdivide this rectangular plot of land into congruent

square pieces. What is the side length of the largest possible square?

3200 m

2400 m

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Multiplying Polynomials:

Monomial × Polynomial

When multiplying a monomial to a polynomial, the monomial is

distributed into the brackets

Ex) 3𝑥(5𝑥2 + 9) 2𝑎𝑏2(14𝑎 − 5𝑎2𝑏3 + 2𝑏)

Ex) Expand the following.

a) ( )23 5 7x x x− b) ( )3 4 35 9 3 2 1x x x x− + −

c) ( )2 3 4 27 3 2 6xy x y x y x+ − + d) ( )4 310 6 2y x y xy− +

12

Polynomial × Polynomial

When multiplying one polynomial to another, each term of one

polynomial must be multiplied to each term of the other

polynomial. Once fully expanded look for like terms in order to

simplify the answer.

Ex) (2𝑥2 + 5)(𝑥 − 7)

Ex) (𝑥 − 4)(𝑥 + 12)

Ex) ( )( )22 5 3 8x x x+ + −

13

Ex) Expand and simplify the following.

a) (𝑎 + 8)(2𝑎 − 3)

b) (3𝑥 − 𝑦)(𝑥 + 3𝑦)

c) (𝑥 + 4)(3𝑥2 + 7𝑥 − 10)

d) (𝑥2 + 8𝑥 − 2)(3𝑥2 − 5𝑥 + 6)

14

e) (𝑥 + 3)(2𝑥 − 1)(4𝑥 + 5)

f) ( )( )2 2 23 5 7 2xy x xy y+ − +

g) ( ) ( )2 33 2 5 3x x x+ + −

h) ( )( )2 27 1 2 3 5a a a a+ − +

15

i) ( )( ) ( )( )24 6 2 3 5x x x x+ − + + −

j) ( ) ( )( )24 1 7 3 2 5x x x− − + −

k) ( )( ) ( )2

3 4 2 5 2x y x y x− + − +

16

Multiplying Polynomials Assignment:

1) Expand and simplify the following.

a) ( )( )2 22 3 1t t t t+ + + + b) ( )( )22 3 4 7w w w+ + +

c) ( )( )2 3x y x y+ + d) ( )( )24 3 3 4 1x x x+ − +

e) ( )( )5 4 5 4x x+ − f) ( )2

3 6t +

g) ( )( )2 5 3 4a b a b+ + + h) ( )( )2 9 2 2 1b b b+ − −

17

i) ( )( )2 22 2 1 6 3d d d d+ + − + j) ( )( )24 4 16x x x− + +

k) ( )( )2 4a b a b c− + − l) ( )( )2 25 3 4a a a− + −

2) Expand and simplify the following.

a) ( )( )2 2 1 3 2a a a− + b) ( )( )25 2 1 4 3x x x− −

c) ( )( )2 2b b c b c− + d) ( )( )( )3 4 5 2 8x x x+ − +

18

e) ( )( )2

5 3 2 7a a− − f) ( )3

2 5x y+

g) ( )3

x y z+ −

3) Expand and simplify the following.

a) ( )( ) ( )( )3 5 2 2 3 7 6x x x x+ + + + +

b) ( )( ) ( )( )2 3 5 4 4 3 7x x x x+ + + − −

19

c) ( )( ) ( )( )4 5 3 2 3 2 2 1y y y y− + − − +

d) ( )( ) ( )( )3 4 4 5 2 3 6m m n m m n+ − + − −

e) ( ) ( )( )2

3 2 2 6 3 1x x x− − − −

f) ( )( ) ( )2

2 1 4 3 2a a a+ − − −

20

4) Determine a simplified polynomial expression that represents the area of the

shaded region for each figure given below.

a)

5 8x +

6 2x +

b)

2 2x −

1x +

5x +

3x

x

2x −

21

Common Factors:

When factoring a polynomial, identify the greatest common

factor (largest term that each term in the polynomial can be

evenly divided by) and divide this out from each term. The

result is written as the product of the greatest common factor

and a polynomial.

Ex) Factor the following by removing a common factor.

15 5x − 3 512 18 24x x x+ −

2 4 3 4 220 15 10a b a b a b− +

Ex) Factor the following.

a) 26 4a a+ b) 2 3 48 14xy x y−

22

c) 25 10 5t t− − d) 5 2 48 12 20a a a+ −

e) 3 2 2 212 20 16x y xy x y− + f) 3 2 2 4 510 21x yz x y z+

g) 2 2 3 3 2 3100 125 300 150a bc ac a bc b c+ − −

h) 2 2 2 3 4 432 16 8x y z xy z xy z+ −

i) 3 2 2 2 3 2 2 25 6 4 8x y x yz y z x y z− + +

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Common Factors Assignment:

1) Factor the following by removing a common factor.

a) 5 10y + b) 2 34 14x x+ c) 2 3 3 29 12a b a b−

d) 23 12 6x x+ − e) 210 6 12a a− −

f) 4 36 7 8x x x+ − g) 2 4 33 13 12a a a− − −

h) 2 3 2 324 30 12x y x y+ − i) 2 2 2 3 4 214 35 21ab c a bc a b c− +

j) 4 3 29 3 8a b a b a b+ − k)

3 3 2 2 37 14 21x y x y xy+ −

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2) Simplify each expression by combining like terms, then factor.

a) 2 26 7 2 3x x x x+ − − − + b) 2 212 24 3 4 13a a a− − + −

c) 3 2 2 37 5 2 12y y y y y y− − + − − − d) 2 2 2 2 2 27 11 18 5 5xy x y x y x y xy− + + +

25

Factoring by Grouping:

Factoring by grouping can be used to factor polynomials with 4

terms.

Ex) 22 8 3 12x x x+ + + 22 8 3 12x x x= + + + → Group the terms

( ) ( )2 4 3 4x x x= + + + → Remove a common factor

from each group. The

bracketed sections should be

the same.

( )( )4 2 3x x= + + → Factor out the common

bracketed binomial from each

group.

Ex) Factor the following by grouping.

a) 22 10 3 15m m m− + − b) 210 5 2a a b b− + −

c) 2 23 2 6x xy xy y− − + d) 4 2 8ac ad bc bd+ + +

26

e) 2 8 3 24x x x− + − f) 3 27 7x x x− + −

g) 9 3 3ac a bc b+ + + h) 25 30 15 90x x x− − +

i) 2 4 2 8 3 12x x xy y x+ + + − −

j) 22 2 2 2ac ab a bc b b− + + − +

27

Factoring by Grouping Assignment:

1) Factor the following using the method of grouping.

a) 210 2 15 3x x x+ + + b) 2 4 8ab a b+ − −

c) 26 3 4 2x x x− − + d) 29 15 15 25x x x+ + +

e) 2 28 4 6 3a ab ab b+ + + f) 2 2 2 27 4 28x x y y+ + +

g) 225 10 10 4x x x− − + h)

4 2 24 8 5 10a a a− + −

28

2) Factor the following by first removing a common factor and then using the

method of grouping.

a) 23 21 6 42x x x+ − − b) 2 250 10 150 30a ab b+ − −

c) 3 2 22 5 12 30x x x x+ + + d)

2 3 2 2 3 29 27 3a b a b ab ab− + −

e) 4 3 3 218 6 6 2x x x x− − + f) 15 20 30 40ac ad bc bd+ + +

g) 2 2 2 218 27 12 18x y x y xy xy+ − − h)

230 6 75 15a b ab ab b− + − +

29

3) Factor the following using the method of grouping.

a) 3 2 26 2 15 5 12 4x x x x x− + − − +

b) 2 25 4 2 5 4 2a ab a ab b b+ − − − +

c) 212 10 6 5 42 35x x xy y x− + − − +

30

Factoring by Decomposition:

Factoring using the method of decomposition is typically used to

factor trinomials. Essentially it is a method that turns a trinomial

(3 terms) into a polynomial of 4 terms that can then be factored

by grouping.

Ex) 23 17 10x x+ + _____ _____ 17

_____ _____ 30

+ =

=

→ Find two numbers that add

up to the middle coefficient

(17) and multiply to the

first times the last

coefficient (30).

23 15 2 10x x x+ + + → Use these numbers to break

up the middle term.

( ) ( )3 5 2 5x x x+ + + → Complete using the method

( )( )5 3 2x x+ + of factoring by grouping.

Ex) Factor the following using the method of decomposition.

a) 26 7 20a a− − b) 22 21a a+ −

31

c) 2 28 14 3x xy y+ + d) 4 211 30x x+ +

e) 28 34 9a a+ − f) 22 5 12x x− −

g) 2 2 48x x+ − h) 215 16 4x x− +

32

i) 212 13 4x x+ − j) 4 5 36a x+ −

k) 24 9x − l) 210 21 10a a+ −

m) 290 45 10x x+ − n) 2 81x −

33

Factoring by Decomposition Assignment:

Factor the following using the method of decomposition.

1) 214 23 3x x− + 2) 2 210 3a ab b+ −

3) 23 13 10a a+ − 4) 2 13 30x x+ −

5) 249 1x − 6)

2 22 9 5a ab b− −

7) 211 8 3a a+ − 8)

2 10 75x x− −

34

9) 29 18 8x x− + 10) 4 25t −

11) 25 17 12x x− − 12) 6 314 48x x+ +

13) 2 212 5 2a ab b− − 14)

26 23 15x x+ +

15) 2100 9a − 16)

2

2 22 60x x+ +

35

Factoring Strategies:

Factor the following.

228 35 42a a− − 230 95 75xy xy y− +

3 2 2 3 412 12 72a b a b ab+ − 290 39 30x x+ −

Strategy:

36

Ex) Factor the following.

2 10 21x x+ + 2 6 16a a+ −

2 3 40x x− − 2 5 36x x+ −

2 16 48t t− + 2 6 40x x− −

Strategy:

37

Ex) Factor the following.

2 16x − 24 25a −

29 49t − 2 100x −

2 64x − 2100 121x −

Strategy:

38

Ex) Factor the following.

a) 2 4 12x x− − b) 22 3 14x x+ −

c) 28 37 15x x+ − d) 236 49x −

e) 24 2 10 5ab a b b+ + + f) 2 11 24b b+ +

g) 2 24 81a b− h) 2 12 32x x+ −

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Factoring Strategies Assignment:

Fully factor the following polynomials.

1) 2 4 32x x+ − 2) 212 23 10x x− +

3) 3 26 72a a a− − 4) 6 4 6 4ab a b+ + +

5) 2 216a b− 6) 28 26 7x x− −

7) 3 22 6 12x x x+ − − 8)

2 15 54t t− +

40

9) 4 16x − 10) 3 2 2 2 23 15 75x y x y xy+ +

11) 212 7 10a a+ − 12) 22 22 60b b+ +

13) 2 64a − 14) 2 10 24x y xy y− −

15) 2 10 16x x− + 16) 2 4 25x y −

41

17) 3 2 3 2 215 20 40a b ab a b− + 18) 224 14 3t t− −

19) 2 14 49x − + 20) 24 28 72b b+ −

21) 2 24 9x y− 22) 24 12 9x x+ +

23) 4 213 36x x− + 24) ( )2

3 25x + −

42

Answers

Prime Factorization / Greatest Common Factor / Lowest Common Multiple

Assignment:

1. a) 6, 12. 18, 24, 30, 33 b) 13, 26, 39, 52, 65, 78

c) 22, 44, 66, 88, 110, 132 d) 31, 62, 93, 124, 155, 186

e) 45, 90, 135, 180, 225, 270 f) 27, 54, 81, 108, 135, 162

2. a) 2 2 2 5 b) 3 5 5 c) 3 3 3 3 d) 2 2 2 3 5

e) 2 2 5 7 f) 2 3 3 3 3 g) 3 3 5 h) 2 2 2 2 2 5

i) 2 2 2 2 2 3

4. a) 3 b) 8 c) 8 d) 4 e) 32 f) 20 g) 5 h) 40 i) 42

j) 4

5. a) 84 b) 315 c) 195 d) 798 e) 1440 f) 364 g) 3420

h) 5280 i) 900 j) 1080

7. a) 37

65 b)

17

19 c)

13

18 d)

42

61 e)

119

110 f)

33

12

8. 800 m

Multiplying Polynomials Assignment:

1. a) 4 3 24 6 7 2t t t t+ + + + b)

3 22 11 26 21w w w+ + + c) 2 26 5x xy y+ +

d) 3 212 7 8 3x x x− − + e)

225 16x − f) 29 36 36t t+ +

g) 2 22 13 20 3 12a ab b a b+ + + + h)

3 22 117 13 2b b b+ − +

i) 4 3 22 10 5 3d d d− − + j) 3 64x − k)

2 22 2 4a ab ac b bc+ − − +

l) 4 3 23 9 15 20a a a a+ − − +

2. a) 3 212 2 4a a a+ − b) 4 3 240 50 15x x x− + c) 3 2 24 2 2b b c bc+ −

d) 3 26 2 128 160x x x+ − − e) 3 220 152 329 147a a a− + −

f) 3 2 2 38 60 150 125x x y xy y+ + +

g) 3 2 2 2 2 3 2 2 33 3 3 3 6 3 3x x y x z xy xz xyz y y z yz z+ − + + − + − + −

3. a) 29 41 52x x+ + b)

213 4 40x x+ + c) 26 6 8y y− −

d) 218 42 2 4m mn m n− − − e)

23 8 2x x+ − f) 27 2 7a a+ −

4. a) 227 43 16x x+ + b) 2 2 2x x+ −

43

Common Factors Assignment:

1. a) ( )5 2y + b) ( )22 2 7x x+ c) ( )2 23 3 4a b b a− d) ( )23 4 2x x− −

e) ( )22 5 3 6a a− − f) ( )3 26 7 8x x x+ − g) ( )2 23 13 12a a a− + +

h) ( )2 3 2 36 4 5 2x y x y+ − i) ( )2 37 2 5 3abc b ac a b c− +

j) ( )2 29 3 8a b a a+ − k) ( )2 27 2 3xy x y x y+ −

2. a) ( )4 1x − b) ( )28 2 3 2a a− − c) ( )22 4 3 5y y y− + +

d) ( )6 2 3xy y x xy− +

Factoring by Grouping Assignment:

1. a) ( )( )5 1 2 3x x+ + b) ( )( )2 4b a+ − c) ( )( )2 1 3 2x x− −

d) ( )2

3 5x + e) ( )( )2 4 3a b a b+ + f) ( )( )2 27 4y x+ + g) ( )2

5 2x −

h) ( )( )2 22 4 5a a− +

2. a) ( )( )3 7 2x x+ − b) ( )( )210 1 5 3b a+ − c) ( )( )2 5 6x x x+ +

d) ( )( )2 3 9ab b a b− + e) ( )222 3 1x x − f) ( )( )5 3 4 2c d a b+ +

g) ( )( )3 2 3 3 2xy y x+ − h) ( )( )3 5 1 2 5b a a− − +

3. a) ( )( )23 1 2 5 4x x x− + − b) ( )( )5 4 2a b a b+ − −

c) ( )( )6 5 2 7x x y− + −

Factoring by Decomposition Assignment:

1. ( )( )7 1 2 3x x− − 2. ( )( )5 2a b a b− + 3. ( )( )3 2 5a a+ −

4. ( )( )15 2x x+ − 5. ( )( )7 1 7 1x x− + 6. ( )( )2 5a b a b+ −

7. ( )( )11 3 1a a− + 8. ( )( )15 5x x− + 9. ( )( )3 4 3 2x x− −

10. ( )( )2 25 5t t− + 11. ( )( )5 3 4x x+ − 12. ( )( )3 36 8x x+ +

13. ( )( )3 2 4a b a b− + 14. ( )( )6 5 3x x+ + 15. ( )( )10 3 10 3a a− +

16. ( )( )2 5 6x x+ +

44

Factoring Strategies Assignment:

1. ( )( )8 4x x+ − 2. ( )( )3 2 4 5x x− − 3. ( )( )12 6a a a− +

4. ( )( )2 3 2 1b a+ + 5. ( )( )4 4a b a b− + 6. ( )( )4 1 2 7x x+ −

7. ( )( )22 6x x+ − 8. ( )( )9 6t t− − 9. ( )( )( )2 4 2 2x x x+ + −

10. ( )2 23 5 25xy x x+ + 11. ( )( )4 5 3 2a a+ − 12. ( )( )2 5 6b b+ +

13. ( )( )8 8a a− + 14. ( )( )12 2y x x− + 15. ( )( )8 2x x− −

16. ( )( )2 25 5xy xy− + 17. ( )2 25 3 4 8ab a b a− + 18. ( )( )6 1 4 3t t+ −

19. ( )2

7x − 20. ( )( )4 9 2b b+ − 21. ( )( )2 3 2 3x y x y− +

22. ( )2

2 3x + 23. ( )( )( )( )3 3 2 2x x x x− + − + 24. ( )( )2 8x x− +