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Bell Work 1/10/14
1. Multiply using Box method
(2a – 3b)(2a + 4b)
2. Complete #’s 47 and 49 from page 2 on
HW pages.
Objective
The student will be able to:
factor trinomials with grouping
by AC (MAMA) method.
Factoring Chart
This chart will help you to determine
which method of factoring to use.
Type
Number of Terms
1. GCF
2. Grouping
3. Trinomials
2 or more
4
3
Review: (y + 2)(y + 4)
y2
+4y
+2y
+8
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
y2 + 6y + 8
y
+2
y2
+2y
+4 +4y
+8
y
In this lesson, we will begin with y2 + 6y + 8 as our
problem and finish with (y + 2)(y + 4) as our answer.
Here we go! 1) Factor y2 + 6y + 8
Use your factoring chart.
Do we have a GCF? Nope!
Now we will learn Trinomials! You will set up
a table with the following information.
Product of the first and
last coefficients
Middle
coefficient
The goal is to find two factors in the first column that
add up to the middle term in the second column.
We’ll work it out in the next few slides.
1) Factor
2
y
M
A
+ 6y + 8
Create your MAMA table.
Product of the
first and last
coefficients
Multiply
+8
Add
+6
Middle
coefficient
Here’s your task…
What numbers multiply to +8 and add to +6?
If you cannot figure it out right away, write
the combinations.
1) Factor
2
y
+ 6y + 8
Place the factors in the table.
Multiply
+8
Which has
a sum
of +6?
+1, +8
-1, -8
+2, +4
-2, -4
Add
+6
+9, NO
-9, NO
+6, YES!!
-6, NO
We are going to use these numbers in the next step!
1) Factor y2 + 6y + 8
Multiply
+8
Add
+6
+2, +4 +6, YES!!
Hang with me now! Replace the middle number of
the trinomial with our working numbers from the
MAMA table
y2 + 6y + 8
y2 + 2y + 4y + 8
Now, group the first two terms and the last two
terms.
We have two groups!
(y2 + 2y)(+4y + 8)
Almost done! Find the GCF of each group and factor
it out.
If things are done
right, the parentheses
y(y + 2) +4(y + 2)
should be the same.
Factor out the
GCF’s. Write them
in their own group.
(y + 4)(y + 2)
Tadaaa! There’s your answer…(y + 4)(y + 2)
You can check it by multiplying. Piece of cake, huh?
There is a shortcut for some problems too!
(I’m not showing you that yet…)
M
A
2) Factor x2 – 2x – 63
Create your MAMA table.
Product of the
first and last
coefficients
Signs need to
be different
since number
is negative.
Multiply
-63
-63, 1
-1, 63
-21, 3
-3, 21
-9, 7
-7, 9
Add
-2
-62
62
-18
18
-2
2
Middle
coefficient
Replace the middle term with our working
numbers.
x2 – 2x – 63
x2 – 9x + 7x – 63
Group the terms.
(x2 – 9x) (+ 7x – 63)
Factor out the GCF
x(x – 9) +7(x – 9)
The parentheses are the same! Weeedoggie!
(x + 7)(x – 9)
Bell Work 1/13/14
1. Factor by grouping 16k3 - 4k2p2 - 28kp +7p3
2. Factor using AC method 2x2 + 9x + 10
Here are some hints to help
you choose your factors in the
MAMA table.
1) When the last term is positive, the factors
will have the same sign as the middle term.
2) When the last term is negative, the factors
will have different signs.
M
A
2) Factor 5x2 - 17x + 14
Create your MAMA table.
Product of the
first and last
coefficients
Signs need to
be the same as
the middle
sign since the
product is
positive.
Multiply
+70
-1, -70
-2, -35
-7, -10
Add
-17
-71
-37
-17
Replace the middle term.
5x2 – 7x – 10x + 14
Group the terms.
Middle
coefficient
(5x2 – 7x) (– 10x + 14)
Factor out the GCF
x(5x – 7) -2(5x – 7)
The parentheses are the same! Weeedoggie!
(x – 2)(5x – 7)
Hopefully, these will continue to get easier the
more you do them.
Factor
1.
2.
3.
4.
(x + 2)(x + 1)
(x – 2)(x + 1)
(x + 2)(x – 1)
(x – 2)(x – 1)
2
x
+ 3x + 2
Factor
1.
2.
3.
4.
2
6y
(6y2 – 15y)(+2y – 5)
(2y – 1)(3y – 5)
(2y + 1)(3y – 5)
(2y – 5)(3y + 1)
– 13y – 5
2) Factor 2x2 - 14x + 12
Find the GCF!
2(x2 – 7x + 6)
Now do the MAMA table!
Signs need to
be the same as
the middle
sign since the
product is
positive.
Multiply
+6
Add
-7
-1, -6
-7
-2, -3
-5
Replace the middle term.
2[x2 – x – 6x + 6]
Group the terms.
2[(x2 – x)(– 6x + 6)]
Factor out the GCF
2[x(x – 1) -6(x – 1)]
The parentheses are the same! Weeedoggie!
2(x – 6)(x – 1)
Don’t forget to follow your factoring chart when
doing these problems. Always look for a GCF
first!!
Objective
The student will be able to:
factor quadratic trinomials.
Trial and Error Method
Factoring Chart
This chart will help you to determine
which method of factoring to use.
Type
Number of Terms
1. GCF
2 or more
2. Diff. Of Squares 2
3. Trinomials
3
Review: (y + 2)(y + 4)
Multiply using FOIL or using the
Box Method.
Box Method: y + 4
y y2 +4y
+ 2 +2y +8
Combine like terms.
2
FOIL: y + 4y + 2y + 8
y2 + 6y + 8
1) Factor.
2
y
+ 6y + 8
Put the first and last terms into the box
as shown.
y2
+8
What are the factors of y2?
y and y
Objective
The student will be able to:
factor perfect square trinomials.
Factoring Chart
This chart will help you to determine
which method of factoring to use.
Type
Number of Terms
1. GCF
2. Grouping
3. Trinomials
2 or more
4
3
Review: Multiply (y + 2)2
(y + 2)(y + 2)
Do you remember these?
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
y2
Using the formula,
+2y (y + 2)2 = (y)2 + 2(y)(2) + (2)2
2 = y2 + 4y + 4
(y
+
2)
+2y
+4
Which one is quicker?
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
y2 + 4y + 4
1) Factor x2 + 6x + 9
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = x
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(x + 3)2
Yes, b = 3
3) Is the middle term twice the
You can still
product of the a and b?
factor the other way
but this is quicker!
Yes, 2ab = 2(x)(3) = 6x
Bell work 1/15/14
Factor using any method
1. 16xy2 - 24y2z + 40y2
2. 6y2 - 13y – 5
3. 2r2 + 12r + 18
2) Factor y2 – 16y + 64
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = y
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(y – 8)2
Yes, b = 8
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(y)(8) = 16y
Factor m2 – 12m + 36
1.
2.
3.
4.
(m – 6)(m + 6)
(m – 6)2
(m + 6)2
(m – 18)2
3) Factor 4p2 + 4p + 1
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = 2p
Since all three are true,
2) Is the last term a perfect write your answer!
square?
(2p + 1)2
Yes, b = 1
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(2p)(1) = 4p
4) Factor 25x2 – 110xy + 121y2
Does this fit the form of our
Perfect Square Trinomials
perfect square trinomial?
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
1) Is the first term a perfect
square?
Yes, a = 5x
Since all three are true,
2) Is the last term a perfect
write your answer!
square?
(5x – 11y)2
Yes, b = 11y
3) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(5x)(11y) = 110xy
Factor 9k2 + 12k + 4
1.
2.
3.
4.
(3k + 2)2
(3k – 2)2
(3k + 2)(3k – 2)
I’ve got no
clue…I’m lost!
Bell work 1/16/14
Factor: 24x2 + 22x - 10
Bell work 1/17/14
Factor the following.
1. 75x2 – 12
2. -64 + 4m2
3. 50x - 60 + 10x2
Objective
The student will be able to:
factor using difference of squares.
Factoring Chart
This chart will help you to determine
which method of factoring to use.
Type
Number of Terms
1. GCF
2. Grouping
3. Trinomials
-AC Method, Trial & Error, Perfect Squares
4. Difference of Squares
2 or more
4
3
2
Determine the pattern
1
4
9
16
25
36
…
= 12
= 22
= 32
= 42
= 52
= 62
These are perfect squares!
You should be able to list
the first 15 perfect
squares in 30 seconds…
Perfect squares
1, 4, 9, 16, 25, 36, 49, 64, 81,
100, 121, 144, 169, 196, 225
Review: Multiply (x – 2)(x + 2)
First terms: x2
Outer terms: +2x
Inner terms: -2x
Last terms: -4
Combine like terms.
x2 – 4
Notice the
middle terms
eliminate
each other!
x
-2
x2
-2x
+2 +2x
-4
x
This is called the difference of squares.
Difference of Squares
2
2
a - b = (a - b)(a + b)
or
2
2
a - b = (a + b)(a - b)
The order does not matter!!
4 Steps for factoring
Difference of Squares
1. Are there only 2 terms?
2. Is the first term a perfect square?
3. Is the last term a perfect square?
4. Is there subtraction (difference) in the
problem?
If all of these are true, you can factor
using this method!!!
1. Factor x2 - 25
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
x2 – 25
Two terms? Yes
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
( x + 5 )(x - 5 )
Write your answer!
2. Factor 16x2 - 9
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
16x2 – 9
Two terms? Yes
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
(4x + 3 )(4x - 3 )
Write your answer!
3. Factor 81a2 – 49b2
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
81a2 – 49b2
Two terms? Yes
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
(9a + 7b)(9a - 7b)
Write your answer!
Factor
1.
2.
3.
4.
2
x
(x + y)(x + y)
(x – y)(x + y)
(x + y)(x – y)
(x – y)(x – y)
Remember, the order doesn’t matter!
–
2
y
4. Factor
2
75x
– 12
When factoring, use your factoring table.
Do you have a GCF? Yes! GCF = 3
3(25x2 – 4)
Are the Difference of Squares steps true?
Two terms? Yes
3(25x2 – 4)
1st term a perfect square? Yes
2nd term a perfect square? Yes
Subtraction? Yes
3(5x + 2 )(5x - 2 )
Write your answer!
Factor
1.
2.
3.
4.
2
18c
prime
2(9c2 + 4d2)
2(3c – 2d)(3c + 2d)
2(3c + 2d)(3c + 2d)
You cannot factor using
difference of squares
because there is no
subtraction!
+
2
8d
Factor -64 +
1.
2.
3.
4.
prime
(2m – 8)(2m + 8)
4(-16 + m2)
4(m – 4)(m + 4)
Rewrite the problem as
4m2 – 64 so the
subtraction is in the
middle!
2
4m
1) Factor.
2
y
+ 6y + 8
Place the factors outside the box as shown.
y
y
y2
+8
What are the factors of + 8?
+1 and +8, -1 and -8
+2 and +4, -2 and -4
1) Factor.
2
y
+ 6y + 8
Which box has a sum of + 6y?
y
+1
y
+2
y
2
y
+ 8 + 8y
+y
+8
y
2
y
+ 4 + 4y
+ 2y
+8
The second box works. Write the numbers
on the outside of box for your solution.
1) Factor.
2
y
+ 6y + 8
(y + 2)(y + 4)
Here are some hints to help you choose your
factors.
1) When the last term is positive, the factors
will have the same sign as the middle term.
2) When the last term is negative, the factors
will have different signs.
2) Factor.
2
x
- 2x - 63
Put the first and last terms into the box
as shown.
x2
- 63
What are the factors of x2?
x and x
2) Factor.
2
x
- 2x - 63
Place the factors outside the box as shown.
x
x
2
x
- 63
What are the factors of - 63?
Remember the signs will be different!
2) Factor. x2 - 2x - 63
Use trial and error to find the correct
combination!
x
-3
x
-7
x
x2
-3x
+ 21 +21x - 63
x
x2
+ 9 +9x
-7x
- 63
Do any of these combinations work?
The second one has the wrong sign!
2) Factor. x2 - 2x - 63
Change the signs of the factors!
x
+7
x
-9
2
x
+7x
-9x
- 63
Write your solution.
(x + 7)(x - 9)
Bell Work 1/14/14
On new work sheet on green table
• Complete #’s 10,12 by AC method
• Complete # 14 using Trial and Error method
3) Factor.
2
5x
- 17x + 14
Put the first and last terms into the box
as shown.
5x2
+ 14
What are the factors of 5x2?
5x and x
3) Factor. 5x2 - 17x + 14
5x
x
5x2
+ 14
What are the factors of + 14?
Since the last term is positive, the signs of the
factors are the same! Since the middle term
is negative, the factors must be negative!
3) Factor. 5x2 - 17x + 14
When the coefficient is not 1, you must try
both combinations!
5x
-2
5x
-7
x
-7
5x2
- 2x
x
-35x + 14 - 2
5x2
- 7x
-10x + 14
Do any of these combinations work?
The second one! Write your answer.
3) Factor. 5x2 - 17x + 14
(5x - 7)(x - 2)
It is not the easiest of things to do,
but the more problems you do, the
easier it gets! Trust me!
2
2x
4) Factor
+ 9x + 10
(x + 2)(2x + 5)
5) Factor.
2
6y
- 13y - 5
(2y - 5)(3y + 1)
6) 12x2 + 11x - 5
(4x + 5)(3x - 1)
7) 5x - 6 + x2
2
x + 5x - 6
(x - 1)(x + 6)