Factors and products Chapter 3

3.1 – factors & multiples of whole numbers Chapter 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.

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: 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:

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.

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 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.

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 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.

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.

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 6n + 9 = 3(2n + 3) 8d + 12d2 = 4d(2 + 3d) Factor each binomial. a) 3g + 6 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

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: –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

Pg. 155-156, #7-11, 14, 16, 18. Independent practice

3.4 – modelling trinomials as binomial products Chapter 3

challenge Factor: 24x2y3z2 + 4xy2z3 + 8xy3z4

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.

Pg. 158, #1-4 Independent practice

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

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

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)

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

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

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

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

FOIL WORKSHEET

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.

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

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 ax2 + 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 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)

Factoring by decomposition 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. 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) The red and black represent our two factors.  Factored form is (2h + 9)(2h + 1).

Example: box method Factored: (2k – 7)(k + 5) 6k2 –21k 10k –35 2k –7 k a) 4h2 + 20h + 9 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)

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 ac bc 2c c d ad bd 2d 3a 3b 6 3 ac + bc + ad + bd + 2c + 2d + 3a + 3b + 6

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).

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: (2c – 3)(c + 5) + 3(c – 3)(–3c + 1) (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

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

Independent Practice