Download Factoring means finding the things you multiply together to get a

6/25/2015
Factoring means finding the
things you multiply together to
get a given answer.
You did some work with
factoring in grade school.
For instance, you found the
prime factorization of
numbers.
When adding or subtracting
fractions, you used factors to
find the
least
common
denominator.
You have also found factors of
numbers and their common
factors.
In algebra we mostly care
about factoring polynomials.
• We want to find what you
need to multiply together
to get a given polynomial.
• It’s like you’re playing
Jeopardy with
the distributive
property.
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The most common factoring
problems look like this:
Factor x2 + 12x + 35
Almost all the time we will be
factoring quadratic trinomials.
We need to find the
quantities we can multiply
to get this polynomial.
Factor x2 + 12x + 35
Factor x2 + 12x + 35
The answer will have the
format (x + ___)(x + ___)
To find the numbers that go in
the quantity,
find what you can
multiply to get 35
that adds up to 12
Factor x2 + 12x + 35
multiply to get 35
that adds up to 12
The only numbers that do both
are 7 and 5.
Factor x2 + 12x + 35
multiply to get 35
that adds up to 12
The only numbers that do both
are 7 and 5.
So … (x + 7)(x + 5)
So … (x + 7)(x + 5)
(x + 5)(x + 7) is also OK.
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Factor x2 + 13x + 36
Factor x2 + 13x + 36
+
•
Factor x2 + 13x + 36
+
•
There are lots of ways to get 36,
like 6 • 6, 9 •4, and 12 • 3.
Factor x2 + 13x + 36
+
•
There are lots of ways to get 36,
like 6 • 6, 9 • 4, and 12 • 3.
Only 9 + 4 adds up to 13.
Factor x2 + 13x + 36
+
•
There are lots of ways to get 36,
like 6 • 6, 9 • 4, and 12 • 3.
Only 9 + 4 adds up to 13.
So the answer is (x + 9)(x + 4)
Factor
x2 + 13x + 40
x2 + 10x + 24
x2 + 10x + 9
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Factor
x2 + 13x + 40
Factor x2 – 16x + 48
(x + 8)(x + 5)
x2
+ 10x + 24
(x + 6)(x + 4)
x2
+ 10x + 9
(x + 9)(x + 1)
Factor x2 – 16x + 48
Factor x2 – 16x + 48
The rule is still the same
Multiply to get 48
Add to get -16
The rule is still the same
-12 • -4 = 48
-12 + -4 = -16
Factor x2 – 16x + 48
Factor
x2 – 5x + 6
The rule is still the same
-12 • -4 = 48
-12 + -4 = -16
So it’s (x – 12)(x – 4).
x2 – 16x + 55
x2 – 18x + 32
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Factor x2 – x – 72
Factor
x2 – 5x + 6
(x – 2)(x – 3)
x2
– 16x + 55
(x – 11)(x – 5)
2
x – 18x + 32
(x – 16)(x – 2)
Factor x2 – x – 72
+ •
This time we need both
positive and negative factors
because we’re multiplying to
get -72.
Factor x2 – x – 72
+ •
Consider (x + 9)(x – 8)
and (x – 9)(x + 8)
We also need to add to -1.
Both multiply to -72
Only the 2nd adds to -1
So … It’s (x – 9)(x + 8)
Factor x2 + 5x – 24
Factor x2 + 5x – 24
This time we need to
multiply to -24 and
add to positive 5
(x + 8)(x – 3)
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If you have one positive and
one negative factor, the larger
factor has the same sign as
the middle term in the
trinomial.
x2 + 4x – 21 = (x + 7)(x – 3)
Factor
x2 + 5x – 36
x2 – 4x – 32
x2 + 12x – 28
x2 – 3x – 18 = (x – 6)(x + 3)
Factor
x2 + 5x – 36
Factor x2 + 6x + 9
(x – 4)(x + 9)
x2
– 4x – 32
(x – 8)(x + 4)
x2
+ 12x – 28
(x + 14)(x – 2)
Factor x2 + 6x + 9
+ •
Factor x2 + 6x + 9
+ •
(x + 3)(x + 3)
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Factor
x2
+ 6x + 9
+ •
(x + 3)(x + 3)
Most books would write this
as (x + 3)2
Factor
x2 + 16x + 64
(x +
x2
8)2
Factor
x2 + 16x + 64
x2 – 18x + 81
x2 + 12x + 36
Factor x2 – 49
– 18x + 81
(x – 9)2
x2 + 12x + 36
(x + 6)2
x2
Factor – 49
We need to multiply to get -49
0x
Factor – 49
We need to multiply to get -49
We need to add to get 0
x2
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0x
2
Factor x – 49
We need to multiply to get -49
We need to add to get 0
Factor
x2 – 100
x2 – 1
x2 – 25
It’s (x + 7)(x – 7)
Factor
x2 – 100
(x + 10)(x – 10)
x2
Let’s
try
a
bit
of
everything.
–1
(x – 1)(x + 1)
x2
– 25
(x + 5)(x – 5)
x2 + 8x + 12
x2 + 8x + 12
x2
– x – 20
x2
– 16x + 64
x2
(x + 6)(x + 2)
– x – 20
(x – 5)(x + 4)
x2
– 16x + 64
(x – 8)2
x2 – 12x + 27
x2 – 12x + 27
(x – 3)(x – 9)
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x2 – 4x - 45
x2 – 4x - 45
x2
– 16
x2
+ 2x + 1
x2
– 18x + 72
x2
(x – 9)(x + 5)
– 16
(x – 4)(x + 4)
x2
+ 2x + 1
(x + 1)(x + 1)
x2
– 18x + 72
(x – 12)(x – 6)
Your book also likes problems
like this.
Factor a2 + 2ab – 15b2
Factor a2 + 2ab – 15b2
The rules are still the
same, but the answer will
have both a and b in it.
Factor a2 + 2ab – 15b2
Multiply to get -15
Add up to 2
Factor a2 + 2ab – 15b2
Multiply to get -15
Add up to 2
It’s (a + 5b)(a – 3b)
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Factor
x2 + 6xy + 8y2
Factor
x2 + 6xy + 8y2
n2
n2
(x + 4y)(x + 2y)
– 2np –
35p2
– 2np –
35p2
(n + 5p)(n – 7p)
Your book also likes problems
like this.
Factor x10 + 16x5 + 63
Factor x10 + 16x5 + 63
What’s different this time is
that the first part has x10.
This means the answer will
have the form
(x5 + __)(x5 + __)
Factor x10 + 16x5 + 63
+
•
Factor x4 – 26x2 + 25
completely.
Everything else is the same.
So, the answer is …
(x5 + 9)(x5 + 7)
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Factor x4 – 26x2 + 25
completely.
Factor x4 – 26x2 + 25
completely.
(x2 – 25)(x2 – 1)
Factor x4 – 26x2 + 25
completely.
(x2 – 25)(x2 – 1)
Factor x4 – 26x2 + 25
completely.
(x2 – 25)(x2 – 1)
… BUT, we’re not done.
• Both parts can be factored
again.
(x + 5)(x – 5)(x + 1)(x – 1)
There are two
more things that
can complicate
factoring.
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First … Quadratic coefficients
If there’s a coefficient that
makes the problem look like
ax2 + bx + c
The answer will usually have
the form
(ax + __)(x + __)
ax2 + bx + c
You still want to find numbers
that will multiply to “c”.
ax2 + bx + c
Unfortunately, they WON’T
just add up to b.
(ax + __)(x + __)
ax2 + bx + c
Remember FOIL.
Factor 3x2 + 23x + 14
Outside + Inside
needs to
add to b
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Factor 3x2 + 23x + 14
Factor 3x2 + 23x + 14
The answer will have the form
(3x + __)(x + __)
The answer will have the form
(3x + __)(x + __)
Since 7 • 2 = 14, it might be
(3x + 7)(x + 2)
or (3x + 2)(x + 7)
Factor 3x2 + 23x + 14
Which is right?
(3x + 7)(x + 2)
Factor 3x2 + 23x + 14
Which is right?
(3x + 7)(x + 2)
6 + 7 = 13
(3x + 2)(x + 7)
Check outside + inside
Factor 3x2 + 23x + 14
(3x + 2)(x + 7)
21 + 2 = 23
☺
Factor 5x2 + 2x – 3
The answer is
(3x + 2)(x + 7)
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Factor 5x2 + 2x – 3
Could be (5x + __)(x – __)
or (5x – __)(x + __)
Factor 5x2 + 2x – 3
The numbers at the end
will be 3 and 1
(one + and one –)
Factor 5x2 + 2x – 3
Consider
(5x + 3)(x – 1)
(5x + 1)(x – 3)
(5x – 3)(x + 1)
(5x – 1)(x + 3)
Check outside + inside
Factor 5x2 + 2x – 3
Consider
(5x + 3)(x – 1) -5+3= -2
(5x + 1)(x – 3) -15+1= -14
(5x – 3)(x + 1) 5–3 = 2 ☺
(5x – 1)(x + 3) 15–1 = 14
Check outside + inside
Factor 5x2 + 2x – 3
Factor
2x2 + 19x + 24
The answer is
7x2 – 37x + 10
(5x – 3)(x + 1)
3x2 – x – 10
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Factor
2x2 + 19x + 24
(2x + 3)(x + 8)
2
7x – 37x + 10
(7x – 2)(x – 5)
2
3x – x – 10
(3x + 5)(x – 2)
The other possible
complication is
common
factors.
The answer typically has the
form ___( __ + __ )
Common factor problems
usually involve binomials, like
this one:
Factor
6x7
+
15x6
Factor 6x7 + 15x6
•
•
The common factor goes
outside the parentheses.
Divide the original problem
by the common factor to
get what stays in the
parentheses.
Factor 6x7 + 15x6
To find the common factor…
• Find the biggest number
that goes into both 6 and
15 (the GCF)
• Choose the smaller
exponent
… Here it’s 3x6
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Factor 6x7 + 15x6
Factor 6x7 + 15x6
So our answer has the form
3x6 ( __ + __ )
So our answer has the form
3x6 ( __ + __ )
Now divide both terms by
3x6
• Divide coefficients.
• Subtract exponents.
Factor 6x7 + 15x6
The final answer is …
Factor 12x4y2 – 8xy3
3x6(2x + 5)
Factor 12x4y2 – 8xy3
Common factor is
4xy2
So answer is
4xy2(__ + __)
Factor 12x4y2 – 8xy3
4xy2(3x3 – 4y)
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Factor
7x5 + 21x4
Factor
7x5 + 21x4
7x4(x + 3)
18x3 – 27x4
18x3 – 27x4
9x3(2 – 3x)
30a5b2 + 25a3b3
Factor 2x4 + 16x3 + 30x2
completely.
30a5b2 + 25a3b3
5a3b2(6a2 + 5b)
Factor 2x4 + 16x3 + 30x2
completely.
First take out a common
factor. Here it’s 2x2
Factor 2x4 + 16x3 + 30x
completely.
Factor 2x4 + 16x3 + 30x
completely.
2x2(x2 + 8x + 15)
2x2(x2 + 8x + 15)
= 2x2(x + 5)(x + 3)
Now factor what’s inside the
parentheses.
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