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MFM2P Polynomial Checklist
1
Goals for this unit:
I can apply the distributive law to the product of binomials.
I can complete the following types of factoring; common, difference of squares and simple trinomials.
U1L1 Getting Ready for Polynomials Learning Goal: I know how to collect like terms and apply the distributive law in algebraic expressions.
Note
Assigned Work: Page 278, # 1 - 10
U5L2 Multiplying Binomials Learning Goal: I can apply distributive law successfully when multiplying binomials and determine their product.
Journal 1
Note
Assigned Work: Page 286, (1 - 4) every other, 6, 9, 10, 15
U5L3 Common Factoring Learning Goal: I know what a common factor is and I can divide it out of a polynomial to create a product (multiplication question)
Journal 2
Note
Assigned Work: Page 294, # 1 - 8, 10, 14, 17, 18
U5L4 Factoring Difference of Squares Learning Goal: I can spot a difference of squares expression and turn it back into two brackets.
Journal 3
Note
Assigned Work: Page 302, #1-8, 10, 14, 17, 18
U5L5 Factoring Simple Trinomials Learning Goal: I can take a trinomial of the form x2+bx+c and turn it back into a product of two brackets.
Journal 4
Note
Assigned Work: Page 309, #2 - 4, 7 - 13, 15, 17
U5L6 Factoring Simple Trinomials With Common Factors Learning Goal: I can factor a simple trinomial that has a common factor, by first taking out the common factor.
Journal 5
Note
Assigned Work: Page 311, #16
Review Journal 6
Assigned Work: See Website
MFM 2P U5L1 Getting Ready for Polynomials
Multiply coefficients and add exponents on the variables.
7.0 Get Ready for Polynomials
Parts of a term...
5x2
Names of Polynomials
Monomial : ________________Binomial : _________________Trinomial : _________________Polynomial : ________________
Multiplying Monomials
5x(3x2)
10x3
3x2
Dividing Monomials
Topic : Review of Algebra Concepts
Goal : I know how to collect like terms and apply the distributive law in algebraic expressions.
Divide coefficients and subtract exponents on the variables.
MFM 2P U5L1 Getting Ready for Polynomials
Collecting Like Terms
5x2 + 4x - 3 + 10x - 2x2 - x + 9
Distributive Law
2x(x+3)
Squaring - (multiplying something by itself)
(-2)2 -22 (4x)2
Practice Page 278 #1-10
Multiply the monomial through the brackets.
Add the coefficients on anything term that has the same variables
and exponents.
U5L2 Multiplying Binomials.notebook July 31, 2013
Multiplying Binomials
Today's goal: I can apply distributive law successfully when multiplying binomials and determine their product.
Recall: Distributive Law
1) 3(x + 5) 2) x(2x 5) 3) 3x(2x + 3)
Recall: What is a Binomial?
Multiplying Binomials
Method 1 Distributive Law Method 2 FOIL
(x + 5)(2x 6) (x + 5)(2x 6)
Examples: Simplify
1) (2x 5)(3x + 7) 2) (3x 8)(x 5)
MFM2P U5L3 Common Factoring
7.2 Common FactoringCommon factoring is the process of reversing distributive law.Expand
3x(4x + 1)
What math operation ( +, -, ×, ÷) did you use to expand the problem?
We need to reverse that process.
It isn't always easy to see what we should divide back out of the expression. We need to find the Greatest Common Factor of the terms.
Finding the GCF of MonomialsWhat is the greatest common factor of...
12x2 and 18x3
STEP 1.STEP 1. Determine the GCF of the coefficients _______
STEP 2.STEP 2. Determine variables common to all _______STEP 3.STEP 3. Determine how many variables they each
have in common. Basically you are looking for the lowest exponent. If a variable has no exponent we know it to be one.
Topic : Common factoring
Goal : I know what a common factor is and I can divide it out of a polynomial to create a product (multiplication question)
MFM2P U5L3 Common Factoring
Example 1. Determine the GCF for the following expressions.
a)12x , 24x5
b) 14x3, 35x2
c) 16x4, 24x, 40x6
Reversing Distributive Law (aka Common Factoring)
If you are given an expression, you should always look to see if you can DIVIDE something (the same thing) out of every term.
12x5 + 9x3 - 15x
STEP 1.STEP 1. Determine the GCF of the terms and place it in front of a set of brackets.
STEP 2.STEP 2. Divide the GCF out of each term and put your answer in the brackets.
NOTENOTE When you divide out the common factor, you will always have the same number of terms inside the brackets as you started with.
MFM2P U5L3 Common Factoring
Homework Page 294 #1-8, 10, 14, 17, 18
Example 2. Common factor the following.
a)10x - 5
b) 20x2 + 15x
c) 8x4 + 6x3 - 2x
Example 3. The area of a rectangle is 10x2 - 5x. Find an expression for length and width of the rectangle.
MFM 2P U5L4 Factoring Difference of Squares
7.3 Factoring Difference of SquaresRemember what happened when we expanded questions like these..
(x+3)(x-3)
(2x+5)(2x-5)
(3x-7)(3x+7)
The final answer is called a DIFFERENCE OF SQUARES! Both terms are perfect squares, because they come from multiplying something by itself. And the sign is always going to be a minus, because the
brackets have different signs.
There are three distinct characteristics that make a difference of squares stand out from other factoring questions
1.1.
2.2.
3.3.
Remember some of the more common perfect square numbers...
14 916 25364964 81100121144169196
Topic : Factoring
Goal : I can spot a difference of squares expression and turn it back into two brackets.
MFM 2P U5L4 Factoring Difference of Squares
Example 2. Spot the difference of squares, and then figure out what brackets they came from (this is called FACTORING)
n2 + 4
15p2 - 25
16p3 - 81
A variable is a perfect square as long as the exponent can be split in half (i.e. an even number)
Example 1. Find the square root of each of the following, if it is a perfect square...
1. 9
2. 25x2
3. 36x10
4. 39x3
5. 25x2
6. 36x10
7. 225n8
8. 6x2
9. 64x2y8
MFM 2P U5L4 Factoring Difference of Squares
Homework Page 302 #1-8, 10, 12, 13
Example 3. Sometimes a difference of squares can be disguised by a common factor. If you take the common factor out, then you will see that it can be factored as a difference of squares.
20p2 - 15 27 - 12n232x5 - 18x
MFM 2P U5L5 Factoring Simple Trinomials
7.4 Simple Trinomial Factoring
We learned how to expand binomials already by basically using the distributive law twice....
(x+5)(x-10) =
Now what we want to do is take an expression like x2 - 5x - 50 and go back the other way... figure out the two brackets that it came from.
We'll start where all mathematicians do - we'll look for a pattern.
Topic : Factoring
Goal : I can take a trinomial of the form x2+bx+c and turn it back into a product of two brackets.
MFM 2P U5L5 Factoring Simple Trinomials
Multiplying Binomials with the SAME SIGNS
(x + 4)(x + 3) (x - 4)(x - 3)
When looking for patterns we are going to treat the sign of a term and the number as two separate things.
* the sign of the last term is * the sign of the middle term is
* the last number is a * the middle number is a
Expand and Simplify the following. Be sure to order in descending powers of x.
Multiplying Binomials with DIFFERENT SIGNSExpand and Simplify the following. Be sure to order in descending powers of x.
(x + 4)(x - 3) (x + 3)(x - 4)
When looking for patterns we are going to treat the sign of a term and the number as two separate things.
* the sign of the last term is * the sign of the middle term is
* the last number is a * the middle number is a
always positive the same as in the brackets
product of the constants in the brackets.
sum of the constants in the brackets.
always negative the same as the bigger constant in the brackets
product of the constants in the brackets.
difference of the constants in the brackets.
MFM 2P U5L5 Factoring Simple Trinomials
x2 + bx + cFactoring Simple Trinomials
MFM 2P U5L5 Factoring Simple Trinomials
Examples. Use the factoring flow chart to help you factor the following.
MFM 2P U5L6 Factoring Simple Trinomials with Common Factors
Topic : Factoring
Goal : I can factor a simple trinomial that has a common factor, by first taking out the common factor.
7.4 Factoring Simple Trinomials When There's a Common Factor
Yesterday we learned to factor something like this...
x2 - 20x + 36
Sometimes a simple trinomial can look complex, if it has a common factor.
4x2 + 8x - 96
Once you have taken out the common factor, you can factor the part in the bracket as you would normally factor a simple trinomial.
MFM 2P U5L6 Factoring Simple Trinomials with Common Factors
We will try a few more and then you can move on to the worksheet.
Factor the following expressions fully...
-x2 - 3x + 28
-2x2 - 22x - 60
x3p - 12x2p + 32xp
4x2 - 20x - 56
5p3 - 80p2 + 315p k4 - 2k3 + k2
Homework Page 311 #16 + worksheet(make sure you completed ALL of yesterday's assigned work)
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MFM 2P
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Factoring Trinomials Practice
Factor each completely.
1)
x
2 − 8
x
− 20 2)
5
n
2 + 25
n
3)
m
2 − 8
m
+ 12 4)
6
k
3 − 60
k
2
5)
5
n
2 + 40
n
6)
p
3 − 5
p
2 + 6
p
7)
6
x
3 − 24
x
2 − 72
x
8)
x
2 − 15
x
+ 50
9)
p
2 + 7
p
+ 12 10)
6
v
2 + 48
v
+ 42
11)
3
x
4 + 21
x
3 − 90
x
2 12)
2
n
3 − 4
n
2 − 96
n
13)
k
3 + 5
k
2 14)
x
3 + 2
x
2
15)
2
k
2 − 4
k
− 30 16)
k
3 − 2
k
2 +
k
17)
x
2 + 11
x
+ 18 18)
4
k
2 − 76
k
+ 360
19)
3
n
2 + 3
n
− 36 20)
x
2 + 14
x
+ 40
21)
3
x
2 + 3
x
− 126 22)
4
k
4 − 28
k
3 − 32
k
2
23)
a
2 + 13
a
+ 42 24)
4
n
3 − 16
n
2
25)
x
2 + 4
x
− 60 26)
5
p
3 − 80
p
2 + 315
p
27)
5
n
3 − 25
n
2 − 70
n
28)
v
2 + 16
v
+ 60
29)
6
b
2 − 24
b
+ 18 30)
v
4 − 9
v
3
31)
x
3 − 14
x
2 + 48
x
32)
n
2 + 8
n
− 9
33)
2
x
2 − 4
x
− 6 34)
5
p
2 − 5
p
− 10
35)
6
b
3 − 12
b
2 − 144
b
36)
6
r
2 + 66
r
+ 144
-1-
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37)
p
2 − 13
p
+ 30 38)
b
2 − 11
b
+ 28
39)
x
2 +
x
− 20 40)
2
x
2 + 14
x
+ 20
41)
2
x
3 + 12
x
2 42)
3
p
3 + 3
p
2 − 18
p
43)
4
a
2 + 52
a
+ 144 44)
n
2 + 5
n
− 50
45)
6
r
3 − 24
r
2 − 192
r
46)
r
2 + 4
r
− 12
47)
n
3 + 4
n
2 + 3
n
48)
4
x
4 − 16
x
3 − 180
x
2
49)
5
r
2 + 85
r
+ 360 50)
5
x
2 − 25
x
− 250
51)
k
3 − 3
k
2 52)
5
a
3 − 65
a
2 + 210
a
53)
x
2 + 15
x
+ 56 54)
x
2 −
x
− 12
55)
6
x
2 + 114
x
+ 540 56)
6
n
2 − 12
n
− 288
57)
5
v
2 − 5
v
− 280 58)
6
k
3 − 66
k
2 + 180
k
59)
n
2 +
n
− 56 60)
p
3 +
p
2 − 20
p
61)
a
3 − 10
a
2 + 9
a
62)
2
x
4 − 12
x
3
63)
x
3 + 10
x
2 + 16
x
64)
4
n
2 − 32
n
+ 60
65)
2
k
2 − 18
k
66)
3
k
2 − 24
k
− 27
67)
v
2 + 5
v
+ 4 68)
3
k
3 − 15
k
2 + 12
k
69)
5
x
2 − 25
x
+ 20 70)
m
2 − 4
m
− 21
71)
n
3 + 4
n
2 − 21
n
72)
4
n
2 + 28
n
+ 40
73)
b
3 −
b
2 − 72
b
74)
x
2 + 9
x
75)
b
2 + 6
b
-2-
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Answers to Factoring Trinomials Practice
1)
(
x
+ 2)(
x
− 10) 2)
5
n
(
n
+ 5) 3)
(
m
− 2)(
m
− 6)4)
6
k
2(
k
− 10) 5)
5
n
(
n
+ 8) 6)
p
(
p
− 3)(
p
− 2)7)
6
x
(
x
− 6)(
x
+ 2) 8)
(
x
− 5)(
x
− 10) 9)
(
p
+ 3)(
p
+ 4)10)
6(
v
+ 7)(
v
+ 1) 11)
3
x
2(
x
+ 10)(
x
− 3) 12)
2
n
(
n
− 8)(
n
+ 6)13)
k
2(
k
+ 5) 14)
x
2(
x
+ 2) 15)
2(
k
− 5)(
k
+ 3)16)
k
(
k
− 1)2 17)
(
x
+ 9)(
x
+ 2) 18)
4(
k
− 9)(
k
− 10)19)
3(
n
− 3)(
n
+ 4) 20)
(
x
+ 4)(
x
+ 10) 21)
3(
x
− 6)(
x
+ 7)22)
4
k
2(
k
+ 1)(
k
− 8) 23)
(
a
+ 7)(
a
+ 6) 24)
4
n
2(
n
− 4)25)
(
x
+ 10)(
x
− 6) 26)
5
p
(
p
− 7)(
p
− 9) 27)
5
n
(
n
+ 2)(
n
− 7)28)
(
v
+ 6)(
v
+ 10) 29)
6(
b
− 1)(
b
− 3) 30)
v
3(
v
− 9)31)
x
(
x
− 6)(
x
− 8) 32)
(
n
+ 9)(
n
− 1) 33)
2(
x
+ 1)(
x
− 3)34)
5(
p
− 2)(
p
+ 1) 35)
6
b
(
b
+ 4)(
b
− 6) 36)
6(
r
+ 8)(
r
+ 3)37)
(
p
− 10)(
p
− 3) 38)
(
b
− 7)(
b
− 4) 39)
(
x
− 4)(
x
+ 5)40)
2(
x
+ 2)(
x
+ 5) 41)
2
x
2(
x
+ 6) 42)
3
p
(
p
+ 3)(
p
− 2)43)
4(
a
+ 9)(
a
+ 4) 44)
(
n
− 5)(
n
+ 10) 45)
6
r
(
r
− 8)(
r
+ 4)46)
(
r
− 2)(
r
+ 6) 47)
n
(
n
+ 1)(
n
+ 3) 48)
4
x
2(
x
+ 5)(
x
− 9)49)
5(
r
+ 8)(
r
+ 9) 50)
5(
x
− 10)(
x
+ 5) 51)
k
2(
k
− 3)52)
5
a
(
a
− 6)(
a
− 7) 53)
(
x
+ 7)(
x
+ 8) 54)
(
x
+ 3)(
x
− 4)55)
6(
x
+ 10)(
x
+ 9) 56)
6(
n
− 8)(
n
+ 6) 57)
5(
v
+ 7)(
v
− 8)58)
6
k
(
k
− 5)(
k
− 6) 59)
(
n
+ 8)(
n
− 7) 60)
p
(
p
+ 5)(
p
− 4)61)
a
(
a
− 1)(
a
− 9) 62)
2
x
3(
x
− 6) 63)
x
(
x
+ 2)(
x
+ 8)64)
4(
n
− 3)(
n
− 5) 65)
2
k
(
k
− 9) 66)
3(
k
+ 1)(
k
− 9)67)
(
v
+ 4)(
v
+ 1) 68)
3
k
(
k
− 4)(
k
− 1) 69)
5(
x
− 1)(
x
− 4)70)
(
m
− 7)(
m
+ 3) 71)
n
(
n
+ 7)(
n
− 3) 72)
4(
n
+ 2)(
n
+ 5)73)
b
(
b
− 9)(
b
+ 8) 74)
x
(
x
+ 9) 75)
b
(
b
+ 6)
