Download Factors and Products Student Notes

CHAPTER 3: FACTORS AND PRODUCTS
Approx. 12 classes
1.
Factors and Multiples of Whole Numbers Perfect
Squares, Perfect Cubes and Their Roots – pg. 134-147
Assignment: pg. 140-141 #3-17, 21
pg. 146-147 #4-10, 12, 13, 15, 17
2.
Multiplying Polynomials – pg. 182-187 (1 class)
Assignment: pg. 186-187 #4-15, 17, 18, 19
3.
Common Factors of a Polynomial and Math Lab – pg. 150-158
Assignment: pg. 155-156 #4-12, 14-16, 18, 20, 21
pg. 158 #1-4
4.
Polynomials of the Form x² + 𝒃𝒙 + 𝒄 – pg. 159-167
Assignment: pg. 166-167 #4-21
5.
Polynomials of the Form 𝒂𝒙² + 𝒃𝒙 + 𝒄 – pg. 168-178
Assignment: pg. 177-178 #5-20
6.
Factoring Special Polynomials – pg. 188-195 (1 class)
Assignment: pg. 194-195 #4-8, 10-13, 15, 18, 20
7.
Factoring Worksheet and Checkpoint 1 and 2: pg. 149 and pg.
180-181 (Could use a second day for this section if needed)
Assignment: pg. 149 #1-4, 6-9 pg. 180-181 #1, 2, 5-9 and
Worksheet
8. Chapter Quiz – (1 class)
1
9. Chapter Review – pg. 198-200 (1 class)
Assignment: pg. 198-200 #1-3, 5-14, 18-22, 24-30, 32-35
10. Chapter Exam - (1 class)
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LESSON 1: FACTORS AND MULTIPLES OF WHOLE NUMBERS
Learning Outcome: Learn to determine prime factors, greatest common
factors and least common multiples of whole numbers. Learn to identify
perfect squares and perfect cubes, then determine square roots and cube
roots.
What is a prime number?
With a partner, list 10 prime numbers:
How do you find the factors of a number?
With the same partner list the factors of 36
Prime factorization of a natural number is the number written as a product
of its prime factors.
Every composite number can be expressed as a product of prime factors.
Ex. Write the prime factorization of 2646
Method 1: Factor Tree
2646
9 x 294
Finish it off!!
3
What does Greatest Common Factor Mean(GCF)?
What is the GCF of 12 and 28?
Ex. Determine the GCF of 126 and 144:
What does Least Common Multiple Mean (LCM)?
What is the LCM of 20 and 25?
Ex. Determine the LCM of 28, 42, and 63:
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The edge of a cube is 3 units.
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What is the area of one face of the cube?
Why is this number a perfect square?
What is the volume of the cube?
Why do you think it has this name?
Perfect squares and square roots are related to each other. The number
25 is a perfect square. It is formed by multiplying two factors of 5 together.
5² = 25
The square root of 25 is 5 or √25 = √(5)(5) = √5² = 5
Perfect square is a number that can be expressed as the product of two
equal factors: 16 = 4 x 4 or 4²
A square root is one of two equal factors of a number: √49 = √(7)(7) = 7
Area = 25 square units
5
Given that the area of a
Square is 25m², how could
you determine its side
length?
Perfect cubes and cube roots are related to each other. The number 27 is
a perfect cube. It is formed by multiplying three factors of 3 together.
3³ = 27
3
3
3
The cube root of 27 is 3 or √27 = √(3)(3)(3) = √3³ = 3
Some numbers are both perfect squares and perfect cubes:
64 = (8)(8) = 8²
64 = (4)(4)(4) = 4³
A perfect cube is a number that is the product of three equal factors.
64 = (4)(4)(4) = 4³
A cube root is one of three equal factors of a number.
3
3
√512 = √(8)(8)(8) = 8
Given that the volume of a
cube is 216m³, how could you
determine the side length?
Volume = 216
cubic units
Ex. State whether each of the following number is a perfect square, a
perfect cube, both or neither:
a. 121
b. 729
c. 356
6
Ex. Determine the square root of 1764
Write 1764 as a product of prime factors
= (2)(882)
= (2)(3)(294)
= (2)(3)(2)(147)
= (2)(3)(2)(3)(49)
= (2)(3)(2)(3)(7)(7)
Now rearrange the factors into 2 equal groups:
=(2 x 3 x 7)(2 x 3 x 7)
= 42 x 42
Square root of 1764 is 42.
Ex. Determine the cube root of 2744:
Write 2744 as a product of prime factors:
7
Ex. The uranium that Saskatchewan produces in a year has a volume of
about 512m³. If this volume were made into a single cube, what would be
the dimensions of the cube?
Assignment: pg. 140-141 #3-17, 21
Assignment: pg. 146-147 #4-10, 12, 13, 15, 17
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LESSON 2: MULTIPLYING POLYNOMIALS
Learning Outcome: Learn to extend the strategies for multiplying
binomials to multiplying polynomials.
The distributive property can be used to perform any polynomial
multiplication (FOIL). Each term of one polynomial must be multiplied by
each term of the other polynomial.
Recall what a like term is?
Ex. (3𝑥 − 2𝑦)(4𝑥 + 𝑦)
12𝑥 2 − 5𝑥𝑦 − 2𝑦 2
Ex. −2𝑎(3𝑎2 + 6𝑎 − 5)
Ex. (𝑥 − 3)(𝑥 2 − 5𝑥 + 2)
Ex. Expand and simplify
a. (2r + 5t)²
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Ex.  x  4  x  5    2 x  1
Remember order of
operations. We must first foil
out the square brackets,
simplify then multiply that by
they (2x-1).
Ex.  y  6 y  3   y  2 y  1
  y 2  3 y  6 y  18   y 2  y  2 y  2 
Ex. Expand and simplify
a. (6h + k – 2)(2h – 3) – (4h – 3k)²
Assignment: pg. 186-187 #4-15, 17, 18, 19
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Remember that the
negative sign in front of the
two binomials will change
the values of the
polynomials after
expansion to their opposite.
LESSON 3: COMMON FACTORS OF POLYNOMIALS
Learning Outcome: Learn to model and record factoring a polynomial.
Learn to exploring factoring polynomials with algebra tiles.
With a partner:
Sketch all the ways you can arrange these tiles to form a rectangle.
Beside each sketch, write the multiplication sentence it represents:
1: width of 1, length of 8x + 4
2: width of 2, length of 4x + 2
3: width of 4, length of 2x + 1
Of the three diagrams above, which do you think is simplified to its lowest
terms? Why?
Compare multiplying and factoring in arithmetic and algebra:
In Arithmetic
In Algebra
Multiply factors to form a product:
(4)(7) = 28
Factor a number by writing it as a product
of factors:
28 = (4)(7)
Expand an expression to form a product:
3(2 – 5a) = 6 – 15a
Factor a polynomial by writing it as a
product of factors:
6 – 15a = 3( 2 – 5a)
Factoring and expanding are inverse processes.
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Ex. Factor each binomial: use algebra tiles or factors
a. 3x + 6
b. 16x – 4
c. 8d + 12d²
d. 9𝑥 4 − 27𝑥 2
e. 6 – 12z + 18z²
Ex. Factor the trinomial, then verify your answer by multiplying the
polynomials.
−20𝑐 4 𝑑 − 30𝑐 3 𝑑2 − 25𝑐𝑑
Ex. Factor then verify: 7a²b – 28ab + 14ab²
12
With a partner, obtained a bag of algebra tiles and try to construct a
rectangle using the tiles of the polynomial:
𝑥 2 + 3𝑥 + 2
In order to build a rectangle, knowing the dimensions are helpful.
Put all the needed tiles for the polynomial in front of you the count up the
total tiles needed to complete the rectangle.
What are the potential factors of your total tiles? These are the potential
dimensions of your rectangle.
Try and create a rectangle using the algebra tiles.
Once created (remember there can be no gaps or holes), read off the
length and width.
The length and width correspond to the factored form of 𝑥 2 + 3𝑥 + 2.
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Using the same steps outlined on the previous page, use the following
trinomials to create a rectangle. Once created, provide the length and
width dimensions.
a. 𝑥 2 + 5𝑥 + 4
b. 2𝑥 2 + 5𝑥 + 2
Assignment: pg. 155-156 #4-12, 14-16, 18, 20, 21
Assignment: pg. 158 #1-4
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LESSON 4: POLYNOMIALS OF THE FORM x² +bx + c
Learning Outcome: Learn to use models and algebraic strategies to
multiply binomials and to factor trinomials.
Given the following trinomial:
𝑥 2 + 6𝑥 + 8
With a partner, use a set of algebra tiles to represent the trinomial as a
rectangle. Once completed, write the length and width of the rectangle.
This is called factored form.
Given: (x + 3)(x + 1)
Expand the two binomials
Pay particular attention to the middle step, this step will be the same for the
process of factoring:
ac = 3, b = 4
What two numbers multiply to 4, but add to 3?
x² + x + 3x + 3
(x + 3)(x + 1)
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Follow the steps outline on the previous example. Ex. Expand and simplify:
2. Expand (x + 3)(x – 7)
Now factor c² - 4c – 21 using the process of decompostion:
Factoring and multiplying are inverse processes. We can use this to factor
a trinomial.
Decomposition
When given a trinomial with a coefficient in front of the x 2 that cannot be
factored out, we use the method of decomposition or breaking up the
middle term.
This is done by multiplying the a and c values together, then finding two
numbers which give the product ac and the sum of b. When we find those
two numbers, we decompose b (break it down) to the two numbers and use
the method of grouping.
x² + 5x + 4: ac = 4, b = 5 (4, 1)
𝑥 2 + 𝑥 + 4𝑥 + 4
(𝑥 2 + 𝑥) + (4𝑥 + 4)
𝑥(𝑥 + 1) + 4(𝑥 + 1)
(𝑥 + 1)(𝑥 + 4)
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Ex. Factor:
a. x² + 12x + 20
b. x²-2x-8
c. a²+7a-18
We should always check that the binomial factors are correct by expanding
the product.
Ex. Factor -30 + 7m + m²
First, rewrite the factor as: m² + 7m – 30
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Ex. Factor -5h² - 20h + 60
We will need to factor out a common factor before factoring:
-5(h² + 4h – 12)
Ex. Factor the following:
a. x² - 29x + 28
b. x² + 3xy – 18y²
Assignment: pg. 166-167 #4-21
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LESSON 5: POLYNOMIALS OF THE FORM ax²+bx+c
Learning Outcome: Learn to extend strategies for multiplying binomials
and factoring trinomials.
To multiply two binomials where the coefficients of the variable are not 1,
we use the same strategies as the previous lesson.
Given: 5𝑥 2 + 11𝑥 + 2
Factor using algebra tiles:
Take your factors and expand.
Now take the trinomial and factor using decomposition.
19
Factor the following:
1. 4h² + 20h + 9:
2. Factor 4g² + 11g + 6
Decomposition
When given a trinomial with a coefficient in front of the x 2 that cannot be
factored out, we use the method of decomposition or breaking up the
middle term.
This is done by multiplying the a and c values together, then finding two
numbers which give the product ac and the sum of b. When we find those
two numbers, we decompose b (break it down) to the two numbers and use
the method of grouping.
Ex. 6 x 2  17 x  12
The product ac = 72, the sum of b = 17. What two numbers will multiply
equal 72, and add to 17: 8 and 9
 6 x 2  8 x  9 x  12
  6 x 2  8 x    9 x  12 
 2 x  3x  4   3  3x  4 
  3x  4  2 x  3
20
Ex. 15 x 2  7 x  4
Ex. 6 x 2  13x  8
Ex. 6 x 2  7 xy  2 y 2
Assignment: pg. 177-178 #5-20
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LESSON 6: FACTORING SPECIAL POLYNOMIALS
Learning Outcome: Learn to investigate some special factoring patterns.
With a partner, determine each product:
(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 see in the trinomials and their factors above?
How could you use the patterns to factor these trinomials?
4x² + 20x + 25
9x² - 12x + 4
Write a strategy for factoring polynomials of this type.
22
When given the situation of  x  2 2 . The term “squaring” means multiplying
the number by itself.
Ex.  x  4  =
2
Ex.  3x  5  =
2
Ex.     =
2
General expansion of squared binomials or Perfect Square Trinomials:
 a  b   a 2  2ab  b2
2
 a  b   a 2  2ab  b2
2
Factoring Perfect Square Trinomials:
2
Patterns:  a  b   a 2  2ab  b 2 and
a  b
2
 a 2  2ab  b 2
Factoring PST:
4 x 2  20 x  25 - recognize it first
how recognize: 1st and last term are perfect squares
a  4x2  4x2  2x
c  25  25  5
b  2ab  2  2 x  5   20 x
It works, all the
numbers match and
add up correctly.
Now to factor:
4 x 2  20 x  25
square root the 1st and last terms: a = 2x, c = 5
To write in form pay attention to the sign in front of the b value:
(in this case +)
2
Write in form:  2 x  5 
23
Ex. Factor :
a. 25a 2  20a  4
b. 4a 2  4ab  b2
c. 5c² - 13cd + 6d²
Difference of Squares:
Ex.  x  1 x 1  x2  x  x 1  x2 1
Ex. 3x  53x  5 
Ex.        
Notice that the two terms are
subtracted and each term is
squared (Difference of squares)
General Rule:
 a  b a  b  a2  b2
Factoring Difference of Squares:
Pattern: a2  b2   a  b  a  b 
Factoring DOS:
9 x 2  16 - recognize it first
9x 2
16 - square root the 1st and last terms
3x
4
Write in form: (3x-4)(3x+4)
24
Ex. Factor each:
a. 9 x 2  25
b. 100 y 2  9 x 2
c. 16m2  4n2
d. 162𝑣 4 − 2𝑤 4
2(81𝑣 4 − 𝑤 4 )
Keep going!!
Assignment: pg. 194-195 #4-8, 10-13, 15, 18, 20
Lesson 7: Factoring Worksheet and Checkpoint
Checkpoint 1 and 2: pg. 149 and pg. 180-181 (1 class)
Assignment: pg. 149 #1-4, 6-9 pg. 180-181 #1, 2, 5-9
Chapter Quiz – (1 class)
Chapter Review – pg. 198-200 (1 class)
Assignment: pg. 198-200 #1-3, 5-14, 18-22, 24-30, 32-35
Chapter Exam - (1 class)
25
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