Chapter 3FACTORS AND

PRODUCTS

Chapter 3

3.1 – FACTORS & MULTIPLES OF

WHOLE NUMBERS

PRIME FACTORS

What are prime numbers?

A factor is a number that divides evenly into another number.

What are some factors of 12?1

2

346

12

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

Check to see which of these factors also divide evenly into 198.

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:

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, 18030 = 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.

Independent practice

PG. 140-141, #4, 6, 8, 9, 11, 13, 16, 19.

Chapter 3

3.2 – PERFECT SQUARES, PERFECT

CUBES & THEIR ROOTS

Find the prime factorization of 1024.

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.

The side length of the square is the square root of the area of the 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 edge length of the cube is the cube root of the volume of the cube.

EXAMPLE

Determine the square root of 1296.

Determine the cube root of 1728.

Find the square root of 1764.

Find the cube root of 2744.

Find the prime factorization:

Find the prime factorization:

EXAMPLE

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

Independent practice

PG. 146-147, #4, 5, 6, 7, 8, 10, 13, 17.

Chapter 3

3.3 – COMMON FACTORS OF A

POLYNOMIAL

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 b) 8d + 12d2

Tiles:a) 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

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

EXAMPLE

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) = –4xyPull out the GCD:

–12x3y – 20xy2 – 16x2y2 = –4xy(3x2 – 5y – 4xy)

Factor: –20c4d – 30c3d2 – 25cd

Independent practice

PG. 155-156, #7-11, 14, 16, 18.

Chapter 3

3.4 – MODELLING TRINOMIALS AS

BINOMIAL PRODUCTS

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 x-tiles and 1-tiles.

Can you spot any patterns? Talk to your partner about it.

Independent practicePG. 158, #1-4

Chapter 3

3.5 – POLYNOMIALS OF THE FORM X2 +

BX + C

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:

Consider: (c + 5)(c + 3)

Arrange algebra tiles with dimensions (c + 5) and (c + 3).

(c + 5)(c + 3) = c2 + 8c + 15

Area model:

Consider: (h + 11)(h + 5)

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

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)

(h + 5)(h + 11) = h2 + 5h + 11h + 55

(h + 5)(h + 11)

= h2 + 11h + 5h + 55= h2 + 16h + 55

EXAMPLE

Expand and simplify:a) (x – 4)(x + 2) b) (8 – b)(3 – b)

a) Method 1: Rectangle diagram

Method 2: FOIL

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

Independent practice

PG. 166-167, #6, 8, 11, 12, 15, 19.

Chapter 3

3.6 – POLYNOMIALS OF THE FORM AX2 +

BX + C

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

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).

Factor:

a) 4h2 + 20h + 9 b) 6k2 – 11k – 35

EXAMPLE: BOX METHOD

Factor:

a) 4h2 + 20h + 9 b) 6k2 – 11k – 35

The box method is another way to factor by decomposition.

1. Put the first term in the upper left box.

2. Put the last term in the bottom right box.

3. Multiply those two numbers together.

4. 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 multiply out to what’s in the boxes.

6k2

–35

6 x –35 = –210

–21k

10k

2k –7

k

5

Factored: (2k – 7)(k + 5)

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

Independent practice

PG. 177-178, #1, 9, 15, 19.

Chapter 33.7 – MULTIPLYING

POLYNOMIALS

MULTIPLYING POLYNOMIALS

Consider the multiplication (a + b + 2)(c + d + 3). Can we draw a rectangle diagram for it?a b 2

c

d

3

ac

ad

3a

bc

bd

3b

2c

2d

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

Independent practice

PG. 186-187, #4, 8, 11, 15, 17, 18, 19.

Chapter 3

3.8 – FACTORING SPECIAL

POLYNOMIALS

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