Transcript Document

1. find the prime factorization of a number.
2. find the greatest common factor (GCF) for
a set of monomials.
SOL: A.2c
Designed by Skip Tyler, Varina High School
number that can only be divided
by only one and itself.
A composite number is a
number greater than one that is
not prime.
Prime or composite?
37
prime
51
composite
1.
2.
3.
4.
Prime
Composite
Both
Neither
84 = 4 • 21
= 2•2•3•7
= 22 • 3 • 7
2) Find the prime factorization of -210.
-210
= -1 • 210
= -1 • 30 • 7
= -1 • 6 • 5 • 7
= -1 • 2 • 3 • 5 • 7
45a2b3 =
b
=
9•5•a•a•b•b•b
3•3•5•a•a•b•b•
=
32 • 5 • a • a • b • b • b
Write the variables without exponents.
1.
2.
3.
4.
3  16
344
2234
22223
the largest number that can divide
into all of the numbers.
4) Find the GCF of 42 and 60.
Write the prime factorization of
each number.
42 = 2 • 3 • 7
60 = 2 • 2 • 3 • 5
What prime factors do the
numbers have in common?
Multiply those numbers.
The GCF is 2 • 3 = 6
6 is the largest number that can
go into 42 and 60!
40a2b = 2 • 2 • 2 • 5 a • a • b
48ab4 = 2 • 2 • 2 • 2 • 3 • a • b • b •
b•b
What do they have in common?
Multiply the factors together.
GCF = 8ab
1.
2.
3.
4.
2
4
8
16
5a
Let’s go one step further…
1) FACTOR 25a2 + 15a.
Find the GCF and divide each term
25a2 + 15a = 5a( ___
5a + ___
3 )
25a 2
5a
15a
5a
Check your answer by distributing.
Find the GCF
6x2
Divide each term by the GCF
3 - ___
2x )
18x2 - 12x3 = 6x2( ___
18 x 2
6x2
12 x 3
6x2
Check your answer by
distributing.
GCF = 28ab
Divide each term by the GCF
2 )
a + 2c
28a2b + 56abc2 = 28ab ( ___
___
28a 2b
28ab
56abc 2
28ab
Check your answer by distributing.
28ab(a + 2c2)
1.
2.
3.
4.
x(20 – 24y)
2x(10x – 12y)
4(5x2 – 6xy)
4x(5x – 6y)
GCF = 7
Divide each term by the GCF
4a2 + ___
3b - ____
5b2c2)
28a2 + 21b - 35b2c2 = 7 ( ___
28a
7
2
21b
7
35b 2 c 2
7
Check your answer by distributing.
7(4a2 + 3b – 5b2c2)
1.
2.
3.
4.
2y2(8x – 12z + 20)
4y2(4x – 6z + 10)
8y2(2x - 3z + 5)
8xy2z(2 – 3 + 5)
Factoring Chart
This chart will help you to determine
which method of factoring to use.
Type
Number of Terms
1. GCF
2. Difference of Squares
2 or more
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
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. Difference of Squares
2
3. Grouping
4
1. Factor 12ac + 21ad + 8bc + 14bd
Do you have a GCF for all 4 terms? No
Group the first 2 terms and the last 2 terms.
(12ac + 21ad) + (8bc + 14bd)
Find the GCF of each group.
3a (4c + 7d) + 2b(4c + 7d)
The parentheses are the same!
(3a + 2b)(4c + 7d)
2. Factor rx + 2ry + kx + 2ky
Check for a GCF: None
You have 4 terms - try factoring by grouping.
(rx + 2ry) + (kx + 2ky)
Find the GCF of each group.
r(x + 2y) + k(x + 2y)
The parentheses are the same!
(r + k)(x + 2y)
3. Factor
2
2x
- 3xz - 2xy + 3yz
Check for a GCF: None
Factor by grouping. Keep a + between the groups.
(2x2 - 3xz) + (- 2xy + 3yz)
Find the GCF of each group.
x(2x – 3z) + y(- 2x + 3z)
The signs are opposite in the parentheses!
Keep-change-change!
x(2x - 3x) - y(2x - 3z)
(x - y)(2x - 3z)
4. Factor
3
16k
-
2
2
4k p
- 28kp +
3
7p
Check for a GCF: None
Factor by grouping. Keep a + between the groups.
(16k3 - 4k2p2 ) + (-28kp + 7p3)
Find the GCF of each group.
4k2(4k - p2) + 7p(-4k + p2)
The signs are opposite in the parentheses!
Keep-change-change!
4k2(4k - p2) - 7p(4k - p2)
(4k2 - 7p)(4k - p2)
Objective
The student will be able to:
factor trinomials with grouping.
SOL: A.2c
Designed by Skip Tyler, Varina High School
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)
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!
Is it a Diff. of Squares problem? No way! 3 terms!
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)
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.
(2x + 10)(x + 1)
(2x + 5)(x + 2)
(2x + 2)(x + 5)
(2x + 1)(x + 10)
2
2x
+ 9x + 10
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!!