1. Factors and products
Chapter 3

2. 3.1 – factors & multiples of whole numbers
Chapter 3

3. A number that isn’t prime is called composite.
Prime factors
What are prime numbers?
A factor is a number that divides evenly into another number.
What are some factors of 12?
Which of these numbers are prime?
 These are the prime factors. 1 12
To find the prime factorization of a number, you write it out as a product of its prime factors. 3 6 4
12 = 2 x 2 x 3 = 22 x 3 2
A number that isn’t prime is called composite.

4. Example: prime factorization
Write the prime factorization of 3300.
Draw a factor tree:
Repeated division:
Try writing the prime factorization of 2646.

5. Example: greatest common factor
Determine the greatest common factor of 138 and 198.
Make a list of both of the factors of 138:
Check to see which of these factors also divide evenly into 198.
138: 1, 2, 3, 6, 23, 46, 69, 138
198 is not divisible by 138, 69, 46, or 23.
It is divisible by 6.
The greatest common factor is 6.
Write the prime factorization for each number:

6. What does multiple mean?
multiples
What does multiple mean?
To find the multiples of a number, you multiply it by 1, 2, 3, 4, 5, 6, etc. For instance, what are the factors of 13?
13, 26, 39, 52, 65, 78, 91, 104, …
For 2 or more natural numbers, we can determine their lowest common multiple.

7. example
Determine the least common multiple of 18, 20, and 30.
Write a list of multiples for each number:
18 = 18, 36, 54, 72, 90, 108, 126, 144, 162, 180
20 = 20, 40, 60, 80, 100, 120, 140, 160, 180
30 = 30, 60, 90, 120, 150, 180
The lowest common multiple for 18, 20, and 30 is 180.
Write the prime factorization of each number, and multiply the greatest power form each list:
Find the lowest common multiple of 28, 42, and 63.

8. example
What is the side length of the smallest square that could be tiled with rectangles that measure 16 cm by 40 cm? Assume the rectangles cannot be cut. Sketch the square and rectangles.
What is the side length of the largest square that could be used to tile a rectangle that measures 16 cm by 40 cm? Assume that the squares cannot be cut. Sketch the rectangle and squares.

9. Pg , #4, 6, 8, 9, 11, 13, 16, 19.
Independent practice

10. 3.2 – perfect squares, perfect cubes & their roots
Chapter 3

11. Find the prime factorization of 1024.

12. Perfect squares and cubes
Any whole number that can be represent-ed as the area of a square with a whole number side length is a perfect square.
Any whole number that can be represent-ed as the volume of a cube with a whole number edge length is a perfect cube.
The side length of the square is the square root of the area of the square.
The edge length of the cube is the cube root of the volume of the cube.

13. example Find the square root of 1764. Find the cube root of 2744.
Determine the square root of 1296.
Find the prime factorization:
Find the square root of 1764.
Determine the cube root of 1728.
Find the prime factorization:
Find the cube root of 2744.

14. example
A cube has volume 4913 cubic inches. What is the surface area of the cube?

16. Pg , #4, 5, 6, 7, 8, 10, 13, 17.
Independent practice

17. 3.3 – common factors of a polynomial
Chapter 3

18. challenge
Expand (use FOIL):
(2x – 2)(x + 4)

19. Algebra tiles Take out tiles that represent 4m + 12
Make as many different rectangles as you can using all of the tiles.

20. Factoring: algebra tiles
When we write a polynomial as a product of factors, we factor the polynomial.
4m + 12 = 4(m + 3) is factored fully because the polynomial doesn’t have any more factors.
 The greatest common factor between 4 and 12 is 4, so we know that the factorization is complete.
Think of factoring as the opposite of multiplication or expansion.

21. example 6n + 9 = 3(2n + 3) 8d + 12d2 = 4d(2 + 3d)
Factor each binomial.
a) 3g b) 8d + 12d2 a)
Tiles:
Look for the greatest common factor:
What’s the GCD for 3 and 6?
 3
6n + 9 = 3(2n + 3) b)
Tiles:
Look for the greatest common factor:
What’s the GCD for 8d and 12d2?
 4d
8d + 12d2 = 4d(2 + 3d)
Try it: Factor 9d + 24d2

22. example
Factor the trinomial 5 – 10z – 5z2.
What’s the greatest common factor of the three terms: 5
–10z
–5z2
They are all divisible by 5.
5 – 10z – 5z2 = 5(1 – 2z – z2)
Divide each term by the greatest common factor.
Check by expanding:
5(1 – 2z – z2) = 5(1) – 5(2z) – 5(z2)
= 5 – 10z – 10z2

23. example Factor: –20c4d – 30c3d2 – 25cd
Factor the trinomial: –12x3y – 20xy2 – 16x2y2
Find the prime factorization of each term:
Identify the common factors.
The greatest common factor is (–2)(2)(x)(y) = –4xy
Pull out the GCD:
 –12x3y – 20xy2 – 16x2y2 = –4xy(3x2 – 5y – 4xy)
Factor: –20c4d – 30c3d2 – 25cd

24. Pg , #7-11, 14, 16, 18.
Independent practice

25. 3.4 – modelling trinomials as binomial products
Chapter 3

26. challenge
Factor:
24x2y3z2 + 4xy2z3 + 8xy3z4

27. Can you spot any patterns? Talk to your partner about it.
Algebra tiles
Use 1 x2-tile, and a number of x-tiles and 1-tiles.
Arrange the tiles to form a rectangle (add more tiles if it’s not possible).
Write the multiplication sentence that it represents.
Ex: (x + 2)(x + 3) = x2 + 5x + 6
Repeat with a different number of tiles.
Try again with 2 or more x2-tiles, and any number of x-tiles and 1-tiles.
Can you spot any patterns? Talk to your partner about it.

29. Pg. 158, #1-4
Independent practice

30. 3.5 – polynomials of the form x2 + bx + c
Chapter 3

31. trinomials
What’s the multiplication statement represented by these algebra tiles?

32. Algebra tiles (c + 4)(c + 2) (c + 4)(c + 3) (c + 4)(c + 4)
Draw rectangles that illustrate each product, and write the multiplication statement represented.
(c + 4)(c + 2)
(c + 4)(c + 3)
(c + 4)(c + 4)
(c + 4)(c + 5)

33. Multiplying binomials with positive terms
Algebra Tiles:
Area model:
Consider: (c + 5)(c + 3)
Arrange algebra tiles with dimensions (c + 5) and (c + 3).
Consider: (h + 11)(h + 5)
Sketch a rectangle with dimensions h + 11 and h + 5
(c + 5)(c + 3) = c2 + 8c + 15
(h + 11)(h + 5) = h2 + 5h + 11h + 55
= h2 + 16h + 55

34. challenge
Expand (use FOIL):
(2x – 4)(x + 3)

35. Area models  FoIL (h + 5)(h + 11) = h2 + 11h + 5h + 55
We can see that the product is made up of 4 terms added together. This is the reason that FOIL works.
(h + 5)(h + 11)
= h2 + 11h + 5h + 55
= h2 + 16h + 55
(h + 5)(h + 11) = h2 + 5h + 11h + 55

36. EXAMPLE Expand and simplify: a) (x – 4)(x + 2) b) (8 – b)(3 – b) a)
Method 1: Rectangle diagram
b) Try it!
Method 2: FOIL

37. FOIL WORKSHEET

38. FACTORING = (x + 8)(x + 3) x2 + 12x + 20
Try to form a rectangle using tiles for:
x2 + 12x + 20
x2 + 12x + 20 = (x + 10)(x + 2)
Factoring without algebra tiles:
 10 and 2 add to give 12
 10 and 2 multiply to give 20
x2 + 11x + 24
= (x + 8)(x + 3)
When we’re factoring we need to find two numbers that ADD to give us the middle term, and MULTIPLY to give us the last term.

39. example a) x2 – 8x + 7 b) a2 + 7a – 18 Factor each trinomial:
a) x2 – 2x – 8 b) z2 – 12z + 35
Try it!
a) x2 – 8x + 7 b) a2 + 7a – 18

40. example
Factor: –24 – 5d + d2
When you’re given a trinomial that isn’t in the usual order, first re-arrange the trinomial into descending order.

41. Example: common factors
Factor: –4t2 – 16t + 128

42. Pg , #6, 8, 11, 12, 15, 19.
Independent practice

43. 3.6 – polynomials of the form ax2 + bx + c
Chapter 3

44. Factoring with a leading coefficient
Work with a partner. For which of these trinomials can the algebra tiles be arranged to form a rectangle? For those that can, write the trinomial in factored form.
2x2 + 15x + 7
2x2 + 9x + 10
5x2 + 4x + 4
6x2 + 7x + 2
2x2 + 5x + 2
5x2 + 11x + 2

45. Multiplying Try it: (5e + 4)(2e + 3) (3d + 4)(4d + 2)
Expand: (3d + 4)(4d + 2)
Method 1: Use algebra tiles/area model
Method 2: FOIL
(3d + 4)(4d + 2)
= 12d2 + 6d + 16d + 8
= 12d2 + 22d + 8
Try it: (5e + 4)(2e + 3)

46. Factoring by decomposition
a) 4h2 + 20h b) 6k2 – 11k – 35
If there is a number out front (what we call a “leading coefficient”) that is not a common factor for all three terms, then factoring becomes more complicated.
4h2 + 20h + 9
First, we need to multiply the first and last term.
4 x 9 = 36
The middle term is 20.
We are looking for two numbers that multiply to 36, and add to 20. Make a list of factors!
Factors of 36
Sum of Factors
1, 36 37
2, 18 20
3, 12 15
4, 9 13
6, 6 36

47. Example continued Factor: a) 4h2 + 20h + 9 b) 6k2 – 11k – 35
Our two factors are 2 and 18.
Now, we need to split up the middle term into these two factors:
4h2 + 20h + 9
4h2 + 2h + 18h + 9
We put brackets around the first two terms and the last two terms.
(4h2 + 2h) + (18h + 9)
Now, consider what common factor can come out of each pair of terms.
 2h(2h + 1) + 9(2h + 1) The red and black represent our two factors.
 Factored form is (2h + 9)(2h + 1).

48. Example: box method Factored: (2k – 7)(k + 5) 6k2 –21k 10k –35 2k –7 k
a) 4h2 + 20h b) 6k2 – 11k – 35
The box method is another way to factor by decomposition. 2k –7
Put the first term in the upper left box.
Put the last term in the bottom right box.
Multiply those two numbers together.
Make a list of factors to find two numbers that multiply to –210 and add to –11.
Our two numbers are –21 and 10. Put those numbers in the other two boxes, with the variable.
Look at each column and row, and ask yourself what factors out.
Make sure that the numbers you pick multiply out to what’s in the boxes. k
6k2
–21k 5
10k
–35
6 x –35 = –210
Factored: (2k – 7)(k + 5)

49. Try factoring by decomposition
Try either method of factoring by decomposition to factor these trinomials:
a) 3s2 – 13s – b) 6x2 – 21x + 9

50. Factoring worksheet

51. Pg , #1, 9, 15, 19.
Independent practice

52. 3.7 – multiplying polynomials
Chapter 3

53. Multiplying polynomials
Consider the multiplication (a + b + 2)(c + d + 3). Can we draw a rectangle diagram for it? a b 2 ac bc 2c c d ad bd 2d 3a 3b 6 3
ac + bc + ad + bd + 2c + 2d + 3a + 3b + 6

54. Draw a rectangle diagram to represent (a – b + 2)(c + d – 3).
Try it
Draw a rectangle diagram to represent (a – b + 2)(c + d – 3).

55. example Expand and simplifying:
a) (2h + 5)(h2 + 3h – 4) b) (–3f2 + 3f – 2)(4f2 – f – 6)

56. example
Expand and simplify:
a) (2r + 5t)2 b) (3x – 2y)(4x – 3y + 5)

57. example Expand and simplify: (2c – 3)(c + 5) + 3(c – 3)(–3c + 1)
(3x + y – 1)(2x – 4) – (3x + 2y)2

58. Pg , #4, 8, 11, 15, 17, 18, 19.
Independent practice

59. 3.8 – factoring special polynomials
Chapter 3

60. challenge
Expand:
(x + 2)(x – 4)(2x + 6) – 4(x2 – 2x + 4)(x + 3)

61. Determine each product with a partner
(x + 1)2 (x + 2)2 (x + 3)2
(x – 1)2 (x – 2)2 (x – 3)2
(2x + 1)2 (3x + 1)2 (4x + 1)2
(2x – 1)2 (3x – 1)2 (4x – 1)2
What patterns do you notice?

62. Perfect square trinomial
(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2
(a – b)2 = (a – b)(a – b) = a2 – 2ab + b2

63. example
Factor each trinomial.
a) 4x2 + 12x b) 4 – 20x + 25x2

64. example
Factor each trinomial.
a) 2a2 – 7ab + 3b2 b) 10c2 – cd – 2d2

65. This is called difference of squares.
example
Factor:
a) x2 – 16 b) 4x2 – 25 c) 9x2 – 64y2
This is called difference of squares.

66. Independent Practice