Chapter 3 3.1 FACTORS & MULTIPLES OF WHOLE NUMBERS
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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 3 46 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 = 2 2 x 3 A
number that isnt prime is called composite.
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EXAMPLE: PRIME FACTORIZATION Write the prime factorization of
3300. Draw a factor tree:Repeated division: Try writing the prime
factorization of 2646.
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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:
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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.
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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.
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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.
Chapter 3 3.2 PERFECT SQUARES, PERFECT CUBES & THEIR
ROOTS
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Find the prime factorization of 1024.
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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.
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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:
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EXAMPLE A cube has volume 4913 cubic inches. What is the
surface area of the cube?
ALGEBRA TILES Take out tiles that represent 4m + 12 Make as
many different rectangles as you can using all of the tiles.
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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 doesnt 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.
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EXAMPLE Factor each binomial. a) 3g + 6b) 8d + 12d 2
Tiles:a)Look for the greatest common factor: Whats the GCD for 3
and 6? 3 6n + 9 = 3(2n + 3) b) Tiles:Look for the greatest common
factor: Whats the GCD for 8d and 12d 2 ? 4d 8d + 12d 2 = 4d(2 + 3d)
Try it: Factor 9d + 24d 2
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EXAMPLE Factor the trinomial 5 10z 5z 2. Whats the greatest
common factor of the three terms: 5 10z 5z 2 They are all divisible
by 5. 5 10z 5z 2 = 5(1 2z z 2 ) Divide each term by the greatest
common factor. Check by expanding: 5(1 2z z 2 ) = 5(1) 5(2z) 5(z 2
) = 5 10z 10z 2
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EXAMPLE Factor the trinomial: 12x 3 y 20xy 2 16x 2 y 2 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: 12x
3 y 20xy 2 16x 2 y 2 = 4xy(3x 2 5y 4xy) Factor: 20c 4 d 30c 3 d 2
25cd
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Independent practice PG. 155-156, #7-11, 14, 16, 18.
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Chapter 3 3.4 MODELLING TRINOMIALS AS BINOMIAL PRODUCTS
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CHALLENGE Factor: 24x 2 y 3 z 2 + 4xy 2 z 3 + 8xy 3 z 4
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ALGEBRA TILES Use 1 x 2 -tile, and a number of x-tiles and
1-tiles. Arrange the tiles to form a rectangle (add more tiles if
its not possible). Write the multiplication sentence that it
represents. Ex: (x + 2)(x + 3) = x 2 + 5x + 6 Repeat with a
different number of tiles. Try again with 2 or more x 2 -tiles, and
any number of x- tiles and 1-tiles. Can you spot any patterns? Talk
to your partner about it.
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Independent practice PG. 158, #1-4
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Chapter 3 3.5 POLYNOMIALS OF THE FORM X 2 + BX + C
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TRINOMIALS Whats the multiplication statement represented by
these algebra tiles?
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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)
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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) = c 2 + 8c + 15 Area model:
Consider: (h + 11)(h + 5) Sketch a rectangle with dimensions h + 11
and h + 5 (h + 11)(h + 5) = h 2 + 5h + 11h + 55 = h 2 + 16h +
55
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CHALLENGE Expand (use FOIL): (2x 4)(x + 3)
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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) = h 2 + 5h + 11h + 55 (h + 5)(h + 11) = h 2 +
11h + 5h + 55 = h 2 + 16h + 55
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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!
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FOIL WORKSHEET
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FACTORING Try to form a rectangle using tiles for: x 2 + 12x +
20 x 2 + 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
were factoring we need to find two numbers that ADD to give us the
middle term, and MULTIPLY to give us the last term. x 2 + 11x + 24
= (x + 8)(x + 3)
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EXAMPLE Factor each trinomial: a) x 2 2x 8b) z 2 12z + 35 Try
it! a) x 2 8x + 7b) a 2 + 7a 18
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EXAMPLE Factor: 24 5d + d 2 When youre given a trinomial that
isnt in the usual order, first re-arrange the trinomial into
descending order.
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EXAMPLE: COMMON FACTORS Factor: 4t 2 16t + 128
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Independent practice PG. 166-167, #6, 8, 11, 12, 15, 19.
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Chapter 3 3.6 POLYNOMIALS OF THE FORM AX 2 + BX + C
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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. 2x 2 + 15x + 72x 2 + 9x + 10 5x 2 + 4x + 46x 2 + 7x + 2 2x 2
+ 5x + 25x 2 + 11x + 2
FACTORING BY DECOMPOSITION Factor: a) 4h 2 + 20h + 9b) 6k 2 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)4h 2 + 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 36Sum of Factors 1, 3637 2,
1820 3, 1215 4, 913 6, 636
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EXAMPLE CONTINUED Our two factors are 2 and 18. Now, we need to
split up the middle term into these two factors: 4h 2 + 20h + 9 4h
2 + 2h + 18h + 9 We put brackets around the first two terms and the
last two terms. (4h 2 + 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) 4h 2 + 20h + 9b) 6k 2 11k 35
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EXAMPLE: BOX METHOD Factor: a) 4h 2 + 20h + 9b) 6k 2 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 whats in the boxes. 6k 2 35 6 x 35 = 210 21k 10k 2k
7 k 5 Factored: (2k 7)(k + 5)
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TRY FACTORING BY DECOMPOSITION Try either method of factoring
by decomposition to factor these trinomials: a) 3s 2 13s 10 b) 6x 2
21x + 9
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FACTORING WORKSHEET
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Independent practice PG. 177-178, #1, 9, 15, 19.
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Chapter 3 3.7 MULTIPLYING POLYNOMIALS
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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
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TRY IT Draw a rectangle diagram to represent (a b + 2)(c + d
3).
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EXAMPLE Expand and simplifying: a) (2h + 5)(h 2 + 3h 4) b) (3f
2 + 3f 2)(4f 2 f 6)
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EXAMPLE Expand and simplify: a) (2r + 5t) 2 b) (3x 2y)(4x 3y +
5)
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EXAMPLE Expand and simplify: a)(2c 3)(c + 5) + 3(c 3)(3c + 1)
b)(3x + y 1)(2x 4) (3x + 2y) 2