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FACTORS AND
PRODUCTS
Chapter 3
3.1 – FACTORS &
MULTIPLES OF WHOLE
NUMBERS
Chapter 3
PRIME FACTORS
What are prime numbers?
A factor is a number that divides evenly into another number.
What are some
factors of 12?
1
6
3
2
12
4
Which of these numbers are prime?
 These are the prime factors.
To find the prime factorization of a number, you
write it out as a product of its prime factors.
12 = 2 x 2 x 3 = 22 x 3
A number that isn’t prime is called composite.
EXAMPLE: PRIME FACTORIZATION
Write the prime factorization of 3300.
Draw a factor tree:
Repeated division:
Try writing the prime
factorization of 2646.
EXAMPLE: GREATEST COMMON FACTOR
Determine the greatest common factor of 138 and 198.
Make a list of both of the factors of 138:
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:
Check to see which of these
factors also divide evenly
into 198.
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.
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.
EXAMPLE
a) 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.
b) 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.
PG. 140-141, #4, 6, 8, 9,
11, 13, 16, 19.
Independent
practice
3.2 – PERFECT
SQUARES, PERFECT
CUBES & THEIR ROOTS
Chapter 3
Find the prime
factorization of 1024.
PERFECT SQUARES AND CUBES
Any whole number that can be represented as the area of a square with a whole
number side length is a perfect square.
Any whole number that can be represented 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.
EXAMPLE
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.
EXAMPLE
A cube has volume 4913 cubic inches. What is the surface area of the cube?
PG. 146-147, #4, 5, 6, 7,
8, 10, 13, 17.
Independent
practice
3.3 – COMMON
FACTORS OF A
POLYNOMIAL
Chapter 3
CHALLENGE
Expand (use FOIL):
(2x – 2)(x + 4)
ALGEBRA TILES
Take out tiles that represent 4m + 12
Make as many different rectangles
as you can using all of the tiles.
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.
EXAMPLE
Factor each binomial.
a) 3g + 6
a) Tiles:
b) 8d + 12d2
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
Try it: Factor 9d + 24d2
8d + 12d2 = 4d(2 + 3d)
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)
Check by expanding:
5(1 – 2z – z2) = 5(1) – 5(2z) – 5(z2)
= 5 – 10z – 10z2
Divide each term by the
greatest common factor.
EXAMPLE
Factor the trinomial: –12x3y – 20xy2 – 16x2y2
Find the prime factorization of each term:
-12x3 y = -2 × 2 × 3× x × x× x × y
-20xy2 = -2 × 2 × 5 × x × y× y
Identify the common factors.
-16x2 y2 = -2 × 2 × 2 × 2 × x× x× y× y
 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
PG. 155-156, #7-11, 14,
16, 18.
Independent
practice
3.4 – MODELLING
TRINOMIALS AS
BINOMIAL PRODUCTS
Chapter 3
CHALLENGE
Factor:
24x2y3z2 + 4xy2z3 + 8xy3z4
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 xtiles and 1-tiles.
Can you spot any patterns? Talk to your partner about
it.
PG. 158, #1-4
Independent
practice
3.5 – POLYNOMIALS OF
THE FORM X 2 + BX + C
Chapter 3
TRINOMIALS
What’s the multiplication statement represented
by these algebra tiles?
ALGEBRA TILES
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)
MULTIPLYING BINOMIALS WITH POSITIVE TERMS
Algebra Tiles:
Area model:
Consider: (c + 5)(c + 3)
Consider: (h + 11)(h + 5)
Arrange algebra tiles with
dimensions (c + 5) and (c + 3).
(c + 5)(c + 3) = c2 + 8c + 15
Sketch a rectangle with
dimensions h + 11 and h + 5
(h + 11)(h + 5) = h2 + 5h + 11h + 55
= h2 + 16h + 55
CHALLENGE
Expand (use FOIL):
(2x – 4)(x + 3)
AREA MODELS  FOIL
(h + 5)(h + 11)
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
EXAMPLE
Expand and simplify:
a) (x – 4)(x + 2)
a) Method 1: Rectangle diagram
Method 2: FOIL
b) (8 – b)(3 – b)
b) Try it!
FOIL WORKSHEET
FACTORING
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
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.
x2 + 11x + 24
= (x + 8)(x + 3)
EXAMPLE
Factor each trinomial:
a) x2 – 2x – 8
b) z2 – 12z + 35
Try it!
a) x2 – 8x + 7
b) a2 + 7a – 18
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.
EXAMPLE: COMMON FACTORS
Factor:
–4t2 – 16t + 128
PG. 166-167, #6, 8, 11,
12, 15, 19.
Independent
practice
3.6 – POLYNOMIALS OF
THE FORM AX 2 + BX + C
Chapter 3
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
MULTIPLYING
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)
FACTORING BY DECOMPOSITION
Factor:
a) 4h2 + 20h + 9
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.
a) 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
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)
 Factored form is (2h + 9)(2h + 1).
The red and black represent our two factors.
EXAMPLE: BOX METHOD
Factor:
a) 4h2 + 20h + 9
b) 6k2 – 11k – 35
The box method is another way to factor by decomposition.
2k
–7
k
6k2
–
21k
5
10k
–35
6 x –35 = –210
Factored: (2k – 7)(k +
1.
2.
3.
4.
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.
5. Our two numbers are –21 and 10. Put
those numbers in the other two boxes, with
the variable.
6. Look at each column and row, and ask
yourself what factors out.
7. Make sure that the numbers you pick
5) multiply out to what’s in the boxes.
TRY FACTORING BY DECOMPOSITION
Try either method of factoring by decomposition to factor these trinomials:
a) 3s2 – 13s – 10
b) 6x2 – 21x + 9
FACTORING WORKSHEET
PG. 177-178, #1, 9, 15,
19.
Independent
practice
3.7 – MULTIPLYING
POLYNOMIALS
Chapter 3
MULTIPLYING POLYNOMIALS
Consider the multiplication (a + b + 2)(c + d + 3). Can we draw a rectangle diagram
for it?
a
b
2
c
ac
bc
2c
d
ad
bd
2d
3
3a
3b
6
ac + bc + ad + bd + 2c + 2d + 3a + 3b + 6
TRY IT
Draw a rectangle diagram to represent (a – b + 2)(c + d – 3).
EXAMPLE
Expand and simplifying:
a) (2h + 5)(h2 + 3h – 4)
b) (–3f2 + 3f – 2)(4f2 – f – 6)
EXAMPLE
Expand and simplify:
a) (2r + 5t)2
b) (3x – 2y)(4x – 3y + 5)
EXAMPLE
Expand and simplify:
a) (2c – 3)(c + 5) + 3(c – 3)(–3c + 1)
b) (3x + y – 1)(2x – 4) – (3x + 2y)2
PG. 186-187, #4, 8, 11,
15, 17, 18, 19.
Independent
practice
3.8 – FACTORING
SPECIAL POLYNOMIALS
Chapter 3
CHALLENGE
Expand:
(x + 2)(x – 4)(2x + 6) – 4(x2 – 2x + 4)(x + 3)
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?
PERFECT SQUARE TRINOMIAL
(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2
(a – b)2 = (a – b)(a – b) = a2 – 2ab + b2
EXAMPLE
Factor each trinomial.
a) 4x2 + 12x + 9
b) 4 – 20x + 25x2
EXAMPLE
Factor each trinomial.
a) 2a2 – 7ab + 3b2
b) 10c2 – cd – 2d2
EXAMPLE
Factor:
a) x2 – 16
b) 4x2 – 25
c) 9x2 – 64y2
This is called difference of squares.
Independent
Practice