INTRODUCTION TO FACTORING POLYNOMIALS

1
INTRODUCTION TO FACTORING POLYNOMIALS
MSJC San Jacinto Campus Math Center Workshop
Series Janice Levasseur
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Definitions

• Recall Factors of a number are the numbers that
  divide the original number evenly.
• Writing a number as a product of factors is
  called a factorization of the number.
• The prime factorization of a number is the
  factorization of that number written as a product
  of prime numbers.
• Common factors are factors that two or more
  numbers have in common.
• The Greatest Common Factor (GCF) is the largest
  common factor.

3
Ex Find the GCF(24, 40).
Prime factor each number
24
2
12
2
6
2
3
? 24 2223 233
? GCF(24,40)
23 8
40
2
20
2
10
2
5
? 40 2225 235
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• The Greatest Common Factor of terms of a
  polynomial is the largest factor that the
  original terms share
• Ex What is the GCF(7x2, 3x)
• 7x2 7 x x
• 3x 3 x
• The terms share a factor of x
• ? GCF(7x2, 3x) x

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Ex Find the GCF(6a5,3a3,2a2)

• 6a5 23aaaaa
• 3a3 3aaa
• 2a2 2aa

The terms share two factors of a
? GCF(6a5,3a3,2a2) a2
Note The exponent of the variable in the GCF is
the smallest exponent of that variable the terms
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Definitions

• To factor an expression means to write an
  equivalent expression that is a product
• To factor a polynomial means to write the
  polynomial as a product of other polynomials
• A factor that cannot be factored further is said
  to be a prime factor (prime polynomial)
• A polynomial is factored completely if it is
  written as a product of prime polynomials

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To factor a polynomial completely, ask

• Do the terms have a common factor (GCF)?
• Does the polynomial have four terms?
• Is the polynomial a special one?
• Is the polynomial a difference of squares?
• a2 b2
• Is the polynomial a sum/difference of cubes?
• a3 b3 or a3 b3
• Is the trinomial a perfect-square trinomial?
• a2 2ab b2 or a2 2ab b2
• Is the trinomial a product of two binomials?
• Factored completely?

8
Ex Factor 7x2 3x
Think of the Distributive Law a(bc) ab ac
? reverse it ab ac a(b c)
Do the terms share a common factor?
What is the GCF(7x2, 3x)?
Recall GCF(7x2, 3x) x
Factor out
2
?
7
x

3
x

x
( )
Whats left?
x
x
? 7x2 3x x(7x 3)
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Ex Factor 6a5 3a3 2a2
Recall GCF(6a5,3a3,2a2) a2
3
1
6a5 3a3 2a2 a2( - - )
6a3
3a
2
a2
a2
a2
? 6a5 3a3 2a2 a2(6a3 3a 2)
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• Your Turn to Try a Few

11
Ex Factor x(a b) 2(a b)
Always ask first if there is common factor the
terms share . . .
x(a b) 2(a b)
Each term has factor (a b)
? x(a b) 2(a b)
(a b)( )
x
2
(a b)
(a b)
? x(a b) 2(a b) (a b)(x 2)
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Ex Factor a(x 2) 2(2 x)
As with the previous example, is there a common
factor among the terms?
Well, kind of . . . x 2 is close to 2 - x . . .
Hum . . .
Recall (-1)(x 2)
- x 2 2 x
? a(x 2) 2(2 x)
a(x 2) 2((-1)(x 2))
a(x 2) ( 2)(x 2)
a(x 2) 2(x 2)
? a(x 2) 2(x 2)
(x 2)( )
a
2
(x 2)
(x 2)
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Ex Factor b(a 7) 3(7 a)
Common factor among the terms?
Well, kind of . . . a 7 is close to 7 - a
Recall (-1)(a 7)
- a 7 7 a
? b(a 7) 3(7 a)
b(a 7) 3((-1)(a 7))
b(a 7) 3(a 7)
b(a 7) 3(a 7)
? b(a 7) 3(a 7)
(a 7)( )
b
3
(a 7)
(a 7)
14

• Your Turn to Try a Few

15
To factor a polynomial completely, ask

• Do the terms have a common factor (GCF)?
• Does the polynomial have four terms?
• Is the polynomial a special one?
• Is the polynomial a difference of squares?
• a2 b2
• Is the polynomial a sum/difference of cubes?
• a3 b3 or a3 b3
• Is the trinomial a perfect-square trinomial?
• a2 2ab b2 or a2 2ab b2
• Is the trinomial a product of two binomials?
• Factored completely?

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Factor by Grouping

• If the polynomial has four terms, consider factor
  by grouping
• Factor out the GCF from the first two terms
• Factor out the GCF from the second two terms
  (take the negative sign if minus separates the
  first and second groups)
• If factor by grouping is the correct approach,
  there should be a common factor among the groups
• Factor out that GCF
• Check by multiplying using FOIL

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Ex Factor 6a3 3a2 4a 2
Notice 4 terms . . . think two groups 1st two
and 2nd two
Common factor among the 1st two terms?
GCF(6a3, 3a2)
3a2
2a
1
? 6a3 3a2 3a2( )
2a
1
3a2
3a2
Common factor among the 2nd two terms?
GCF(4a, 2)
2
2
1
2a
1
? 4a 2 2( )
2
2
Now put it all together . . .
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6a3 3a2 4a 2
3a2(2a 1) 2(2a 1)
Four terms ? two terms. Is there a common factor?
Each term has factor (2a 1)
(2a 1)( )
3a2
2
3a2(2a 1) 2(2a 1)
(2a 1)
(2a 1)
6a3 3a2 4a 2
(2a 1)(3a2 2)
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Ex Factor 4x2 3xy 12y 16x
Notice 4 terms . . . think two groups 1st two
and 2nd two
Common factor among the 1st two terms?
GCF(4x2, 3xy)
x
4x
3y
? 4x2 3xy x( )
4x
3y
x
x
Common factor among the 2nd two terms?
GCF(-12y, - 16x)
-4
3y
4x
3y
4x
? -12y 16x - 4( )
-4
-4
Now put it all together . . .
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4x2 3xy 12y 16x
x(4x 3y) 4(4x 3y)
Four terms ? two terms. Is there a common factor?
Each term has factor (4x 3y)
(4x 3y)( )
x
4
x(4x 3y) 4(4x 3y)
(4x 3y)
(4x 3y)
4x2 3xy 12y 16x
(4x 3y)(x 4)
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Ex Factor 2ra a2 2r a
Notice 4 terms . . . think two groups 1st two
and 2nd two
Common factor among the 1st two terms?
GCF(2ra, a2)
a
? 2ra a2 a( )
2r
a
a
a
Common factor among the 2nd two terms?
GCF(-2r, - a)
-1
2r
a
? -2r a - 1( )
-1
-1
Now put it all together . . .
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2ra a2 2r a
a(2r a) 1(2r a)
Four terms ? two terms. Is there a common factor?
Each term has factor (2r a)
(2r a)( )
a
1
a(2r a) 1(2r a)
(2r a)
(2r a)
2ra a2 2r a
(2r a)(a 1)
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• Your Turn to Try a Few

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To factor a polynomial completely, ask

• Do the terms have a common factor (GCF)?
• Does the polynomial have four terms?
• Is the polynomial a special one?
• Is the polynomial a difference of squares?
• a2 b2
• Is the trinomial a perfect-square trinomial?
• a2 2ab b2 or a2 2ab b2
• Is the trinomial a product of two binomials?
• Factored completely?

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Special Polynomials

• Is the polynomial a difference of squares?
• a2 b2 (a b)(a b)
• Is the trinomial a perfect-square trinomial?
• a2 2ab b2 (a b)2
• a2 2ab b2 (a b)2

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Ex Factor x2 4
Notice the terms are both perfect squares
? difference of squares
and we have a difference
x2 (x)2
4 (2)2
? x2 4 (x)2 (2)2
(x 2)(x 2)
a2 b2
(a b)(a b)
factors as
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Ex Factor 9p2 16
Notice the terms are both perfect squares
? difference of squares
and we have a difference
9p2 (3p)2
16 (4)2
? 9a2 16 (3p)2 (4)2
(3p 4)(3p 4)
a2 b2
(a b)(a b)
factors as
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Ex Factor y6 25
Notice the terms are both perfect squares
? difference of squares
and we have a difference
y6 (y3)2
25 (5)2
? y6 25 (y3)2 (5)2
(y3 5)(y3 5)
a2 b2
(a b)(a b)
factors as
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Ex Factor 81 x2y2
Notice the terms are both perfect squares
? difference of squares
and we have a difference
81 (9)2
x2y2 (xy)2
? 81 x2y2 (9)2 (xy)2
(9 xy)(9 xy)
a2 b2
(a b)(a b)
factors as
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• Your Turn to Try a Few

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To factor a polynomial completely, ask

• Do the terms have a common factor (GCF)?
• Does the polynomial have four terms?
• Is the polynomial a special one?
• Is the polynomial a difference of squares?
• a2 b2
• Is the polynomial a sum/difference of cubes?
• a3 b3 or a3 b3
• Is the trinomial a perfect-square trinomial?
• a2 2ab b2 or a2 2ab b2
• Is the trinomial a product of two binomials?
• Factored completely?

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FOIL Method of Factoring

• Recall FOIL
• (3x 4)(4x 5) 12x2 15x 16x 20 12x2
  31x 20
• The product of the two binomials is a trinomial
• The constant term is the product of the L terms
• The coefficient of x, b, is the sum of the O I
  products
• The coefficient of x2, a, is the product of the F
  terms

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FOIL Method of Factoring

1. Factor out the GCF, if any
2. For the remaining trinomial, find the F terms (__
   x )(__ x ) ax2
3. Find the L terms ( x __ )( x __ ) c
4. Look for the outer and inner products to sum to
   bx
5. Check the factorization by using FOIL to multiply

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Ex Factor b2 6b 5
1. there is no GCF
2. the lead coefficient is 1 ? (1b )(1b
)
3. Look for factors of 5
1, 5 5, 1
(b 1)(b 5) or (b 5)(b 1)
4. outer-inner product?
(b 1)(b 5) ? 5b b 6b
or (b 5)(b 1) ? b 5b 6b
Either one works ? b2 6b 5 (b 1)(b 5)
5. check (b 1)(b 5)
b2 5b b 5
b2 6b 5
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Ex Factor y2 6y 55
1. there is no GCF
2. the lead coefficient is 1 ? (1y )(1y
)
3. Look for factors of 55
1, -55 5, - 11 11, - 5 55, - 1
(y 1)(y 55) or (y 5)(y - 11) or ( y 11)(y
5) or (y 55)(y 1)
4. outer-inner product?
(y 1)(y - 55) ? -55y y - 54y
(y 5)(y - 11) ? -11y 5y -6y
(y 55)(y - 1) ? -y 55y 54y
(y 11)(y - 5) ? -5y 11y 6y
? y2 6y - 55 (y 11)(y 5)
5. check (y 11)(y 5)
y2 5y 11y - 55
y2 6y 55
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Factor completely 3 Terms

• Always look for a common factor
• immediately take it out to the front of the
  expression all common factors
• show whats left inside ONE set of parenthesis
• Identify the number of terms.
• If there are three terms, and the leading
  coefficient is positive
• find all the factors of the first term, find all
  the factors of the last term
• Within 2 sets of parentheses,
• place the factors from the first term in the
  front of the parentheses
• place the factors from the last term in the back
  of the parentheses
• NEVER put common factors together in one
  parenthesis.
• check the last sign,
• if the sign is plus use the SAME signs, the sign
  of the 2nd term
• if the sign is minus use different signs, one
  plus and one minus
• smile to make sure you get the middle term
• multiply the inner most terms together then
  multiply the outer most terms together, and add
  the two products together.

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Factor completely 2x2 5x 7

• Factors of the first term 1x 2x
• Factors of the last term -1 7 or 1 -7
• (2x 7)(x 1)

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Factor Completely.4x2 83x 60

• Nothing common
• Factors of the first term 1 4 or 2 2
• Factors of the last term 1,6 2,30 3,20 4,15
  5,12 6,10
• Since each pair of factors of the last has an
  even number,we can not use 2 2 from the first
  term
• (4x 3)(1x
  20 )

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Sign Pattern for the Binomials
Trinomial Sign Pattern Binomial Sign Pattern
(
)( )
- (
- )( - )
- - 1
plus and 1 minus
- 1 plus and
1 minus
But as you can tell from the previous example,
the FOIL method of factoring requires a lot of
trial and error (and hence luck!) . . . Better
way?
40

• Your Turn to Try a Few

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ac Method for factoring ax2 bx c

1. Factor out the GCF, if any
2. For the remaining trinomial, multiply ac
3. Look for factors of ac that sum to b
4. Rewrite the bx term as a sum using the factors
   found in step 3
5. Factor by grouping
6. Check by multiplying using FOIL

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Ex Factor 3x2 4x 15
3
4
No
1. Is there a GCF?
?3(-15) - 45
2. Multiply ac ?
a
3
and c
15
3. Factors of -45 that sum to
4
1 45 ? 44
Note although there are more factors of 45,
we dont have to check them since we found what
we were looking for!
3 15 ? 12
5 9 ? 4
4. Rewrite the middle term
3x2 4x 15 3x2 9x 5x 15
Four-term polynomial . . . How should we proceed
to factor?
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Factor by grouping . . . 3x2 9x 5x 15
Common factor among the 1st two terms?
3x
3
x
? 3 x 2 9x 3x( )
3
3x
3x
Common factor among the 2nd two terms?
5
3
? 5 x 15 5( )
x
3
5
5
? 3x2 9x 5x 15 3x(x 3) 5(x 3)
(x 3)( )
3x
5
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Ex Factor 2t2 5t 12
2
12
5
No
1. Is there a GCF?
?2(-12) - 24
2. Multiply ac ?
a
2
and c
12
3. Factors of -24 that sum to
5
1 24 ? 23
Close but wrong sign so reverse it
- 3 8 ? 5
2 12 ? 10
3 8 ? 5
4. Rewrite the middle term
2t2 5t 12 2t2 3t 8t 12
Four-term polynomial . . . Factor by grouping . .
.
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2t2 3t 8t 12
Common factor among the 1st two terms?
t
3
2t
? 2 t 2 3t t( )
3
t
t
Common factor among the 2nd two terms?
4
2
3
? 8 t 12 4( )
2t
3
4
4
? 2t2 3t 8t 12 t(2t 3) 4(2t 3)
(2t 3)( )
t
4
46
Ex Factor 9x4 18x2 8
9
8
18
No
1. Is there a GCF?
?9(8) 72
2. Multiply ac ?
a
9
and c
8
3. Factors of 72 that sum to
18
1 72 ? 73
Bit big ? think bigger factors
3 24 ? 27
6 12 ? 18 ?
4. Rewrite the middle term
9x4 18x2 8 9x4 6x2 12x2 8
Four-term polynomial . . . Factor by grouping . .
.
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9x4 6x2 12x2 8
Common factor among the 1st two terms?
3x2
2
3x2
3x2
? 9x4 6x2 3x2( )
2
3x2
3x2
Common factor among the 2nd two terms?
4
3
3
? 12x2 8 4( )
3x2
2
4
4
? 9x4 6x2 12x2 8 3x2(3x2 2) 4(3x2
2)
(3x2 2)( )
3x2
4
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17 y
Ex Factor 12x2 17xy 6y2
12
6y2
Pick one to be the variable
No, but notice two variables
1. Is there a GCF?
12x2(6y2) 72y2
2. Multiply ac ?
a
12x2
and c
6y2
3. Factors of 72x2y2 that sum to
- 17xy
-1xy -72xy ? -73xy
Each factor need a y, both need to be negative
-6xy -12xy ? -18xy
Too big, think bigger factors
-8xy -9xy ? -17xy ?
4. Rewrite the middle term
12x2 17xy 6y2 12x2 8xy 9xy 6y2
Four-term polynomial . . . Factor by grouping . .
.
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12x2 8xy 9xy 6y2
Common factor among the 1st two terms?
4x
2y
3x
3x
? 12x2 8xy 4x( )
2y
4x
4x
Common factor among the 2nd two terms?
- 3y
3
-2y
? 9xy 6y2 - 3y( )
3x
2y
-3y
-3y
? 12x2 8xy 9xy 6y2 4x(3x 2y) 3y(3x
2y)
(3x 2)( )
4x
3y
50

• Your Turn to Try a Few

51
To factor a polynomial completely, ask

• Do the terms have a common factor (GCF)?
• Does the polynomial have four terms?
• Is the polynomial a special one?
• Is the polynomial a difference of squares?
• a2 b2
• Is the polynomial a sum/difference of cubes?
• a3 b3 or a3 b3
• Is the trinomial a perfect-square trinomial?
• a2 2ab b2 or a2 2ab b2
• Is the trinomial a product of two binomials?
• Factored completely?

52
Ex Factor x3 3x2 4x 12
No
1. Is there a GCF?
2. Notice four terms
? grouping
Common factor among the 1st two terms?
x2
x
x
3
? x3 3x2 x2( )
x2
x2
Common factor among the 2nd two terms?
- 4
3
? 4x 12 4( )
x
3
- 4
- 4
? x3 3x2 - 4x 12 x2(x 3) 4(x 3)
(x 3)( )
x2
4
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Cont we have (x 3)(x2 4)
But are we done?
No. We have to make sure we factor completely.
Is (x 3) prime? ? can x 3 be factored
further?
No . . . It is prime
What about (x2 4)?
Recognize it?
Difference of Squares
x2 (x)2
4 (2)2
? x2 4 (x)2 (2)2
(x 2)(x 2)
Therefore x3 3x2 4x 12 (x 3)(x2 4)
(x
3)(x 2)(x 2)
54

• Your Turn to Try a Few

55
To factor a polynomial completely, ask

• Do the terms have a common factor (GCF)?
• Does the polynomial have four terms?
• Is the polynomial a special one?
• Is the polynomial a difference of squares?
• a2 b2
• Is the polynomial a sum/difference of cubes?
• a3 b3 or a3 b3
• Is the trinomial a perfect-square trinomial?
• a2 2ab b2 or a2 2ab b2
• Is the trinomial a product of two binomials?
• Factored completely?

56
Special Polynomials

• Is the polynomial a sum/difference of cubes?
• a3 b3 (a b)(a2 - ab b2)
• a3 b3 (a - b)(a2 ab b2)

57
Ex Factor 8p3 q3
Notice the terms are both perfect cubes
? difference of cubes
and we have a difference
8p3 (2p)3
q3 (q)3
? 8p3 q3 (2p)3 (q)3
(2p q)((2p)2 (2p)(q) (q)2)
a3 b3
(a b)(a2 ab b2)
factors as
(2p q)(4p2 2pq q2)
58
Ex Factor x3 27y9
Notice the terms are both perfect cubes
? sum of cubes
and we have a sum
x3 (x)3
27y9 (3y3)3
? x3 27y9 (x)3 (3y3)3
(x 3y3)((x)2 - (x)(3y3) (3y3)2)
a3 b3
(a b)(a2 - ab b2)
factors as
(x 3y3)(x2 3xy3 9y6)
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