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Math 100 Intermediate Algebra
FACTORING POLYNOMIALS OF
THE FORM
2
ax  bx  c, with a  1
Chapter 5, section 5
Delaware County Community
College
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before you begin this tutorial. Ask the
instructor to give you one.
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If you have any questions, do not hesitate to ask the teacher.
Prerequisites
(references are to chapter, section and pages in the textbook)
• Use FOIL to multiply two binomials (4.2,
page 351)
• Understand the difference between a
binomial and a trinomial (4.1, page 339)
• Find the greatest common factor of a
polynomial (4.3, pages 360 - 361)
• Factor a polynomial of this type with a = 1
(4.4, pages 367 - 369)
A polynomial having two terms is a binomial .
To multiply two binomials using FOIL :
F L F L
(3x + 2)(5x – 7)
O I I O
F represents the first terms in the two binomials
O represents the outside terms in the two binomials
I represents the inside terms in the two binomials
L represents the last terms in the two binomials.
F
Multiply:
L
F
L
(3x + 2)(5x - 7)
O
I
I
O
First: (3x)(5x) = 15x 2
Outside: (3x)(-7) = -21x
Inside: (2)(5x) = 10x
Last: (2)(-7) = -14
Now add by combining like terms:
15x2 - 21x  10x -14 = 15x2 -11x -14
Rules for Factoring Trinomials:
1. Factor out the greatest common factor.
Always do this first and remember to include
it in your answer.
2. The next step in the process is determined
by the value of the leading coefficient, a.
If a = 1 one process is followed and if a ≠ 1
a different but similar process is followed.
3. If variables are present in both the First
and Last terms in the trinomial, then take
this into account when writing the final
answer.
Factoring a polynomial means to rewrite it as a product.
If the polynomial is a trinomial (three terms) this means
to un-FOIL it, or to write it as a product of two binomials.
First we consider a trinomial with leading coefficient a = 1
Factor x 2  11x  18
The + in this position
means that the signs
in the two binomial
factors will both be +.
The + in this position
means two things:
1.The sign in the two
binomials will be the
same.
2. You add the factors
of 18 to get 11, the middle
coefficient.
Factor
x 2  11x  18
1. List all of the pairs of factors of 18
1, 18 because (1)(18) = 18
2, 9
3, 6
because (2)(9) = 18
because (3)(6) = 18
2. Add the pairs of factors of 18
1 + 18 = 19
2 + 9 = 11
3+6=9
3. The pair of factors that add to give the middle
coefficient 11 is 2 and 9. These are placed in the last
position (L) of the two binomials.
( x + 2 )( x + 9 )
Check by FOILing: x 2  9x  2x  18 = x 2  11x  18
Factor x 2  14x  45
The − in this position
means that both signs
in the two binomial
factors will be −.
The + in this position
means two things.
1. The sign in the two
binomials will be the
same.
2. You add the factors
of 45 to get 14, the middle
coefficient.
Factor
x 2  14x  45
1. List all of the pairs of factors of 45
1, 45 because (1)(45) = 45
3, 15 because (3)(15) = 45
5, 9
because (5)(9) = 45
2. Add the pairs of factors of 45
1 + 45 = 46
3 + 15 = 18
5 + 9 = 14
3. The pair of factors that add to give the middle
coefficient 14 is 5 and 9. These are placed in the
last (L) position in two binomials.
( x − 5 )( x − 9 )
Check by FOILing:
x 2  9x  5x  45 = x 2  14x  45
TO REVIEW:
To factor a trinomial with leading coefficient 1 with the
sign of the third term in the trinomial +:
x 2  bx  c
or
x 2  bx  c
1. List every pair of numbers that multiply to give c.
2. Add each pair of numbers to determine which pair add to give the
middle coefficient, b.
3. The factors of the trinomial will be two binomials, the first (F) term
in each will be x and the sign in each binomial will be the same as
the sign of the middle coefficient, b.
4. The last (L) term in each binomial will be the factors of c
that add to give the middle coefficient, b.
Factor x 2  13x  30
The − in this position
means two things.
The + in this position
means that the sign of
the largest factor of 30
in the two binomial
factors will be +. The
sign in the other binomial
factor will be −.
1. The sign in the two
binomials will be the
different.
2. You subtract the
factors of 30 to get 13,
the middle coefficient.
Factor
x 2  13x  30
1. List all of the pairs of factors of 30
1, 30
because (1)(30) = 30
2, 15
because (2)(15) = 30
3, 10
because (3)(10) = 30
5, 6
because (5)(6) = 30
2. Subtract the pairs of factors of 30
30 − 1 = 29
15 − 2 = 13
10 − 3 = 7
6–5=1
3. The pair of factors that subtract to give the middle
coefficient 13 is 2 and 15
( x − 2 )( x + 15 )
Check by FOILing: x 2  15x  2x  30 = x 2  13x  30
Factor x 2  19x  42
The − in this position
means that the sign of
the largest factor of 42
in the two binomial
factors will be −. The
sign in the other binomial
factor will be +.
The − in this position
means two things.
1. The sign in the two
binomials will be the
different.
2. You subtract the
factors of 42 to get 19,
the middle coefficient.
Factor
x 2  19x  42
1. List all of the pairs of factors of 42
1, 42
because (1)(42) = 42
2, 21
because (2)(21) = 42
3, 14
because (3)(14) = 42
6, 7
because (6)(7) = 42
2. Subtract the pairs of factors of 42
42 − 1 = 41
21 − 2 = 19
14 − 3 = 11
7–6=1
3. The pair of factors that subtract to give the middle
coefficient 19 is 2 and 21
( x + 2 )(x − 21 )
Check by FOILing:
x 2  21x  2x  42 = x 2  19x  42
TO REVIEW:
To factor a trinomial with leading coefficient 1 with the
sign of the third term in the trinomial −:
x 2  bx  c
or
x 2  bx  c
1. List every pair of numbers that multiply to give c.
2. Subtract each pair of numbers to determine which pair subtract
to give the middle coefficient, b.
3. The factors will be two binomials, the first (F) terms will be x and the
sign in the two binomials will be different.
4. The last (L) term in each binomial will be the factors of c
that add to give the middle coefficient, b.
5. The sign of the largest factor of c will be the same as the sign
of the middle coefficient, b.
Rules for Factoring Trinomials:
1. Factor out the greatest common factor.
Always do this first and remember to include it in
your answer.
2. If the remaining trinomial has leading coefficient 1,
proceed as explained earlier in this tutorial.
3. If the remaining trinomial has leading coefficient
not 1, proceed as follows.
4. If variables are present in both the First and Last
terms in the trinomial, then take this into account
when writing the final answer.
The factors of a number are the numbers that
divide it evenly.
To write a trinomial with a ≠ 1 in factored form
consider the following:
2x 2  15x  7
The + in this position
means that the signs in
the two binomial factors
will both be +.
The + in this position
means two things:
1.The sign in the two
binomials will be the
same.
2. You add the product
of the factors of 2 and 7
to get 15, the middle
coefficient.
Remember, when multiplying two binomials
using FOIL you multiply the First terms, the
Outside terms, the Inside terms and the
Last terms.
F
(
O
L F
)(
I I
L
)
O
To factor 2x 2  15x  7
you need to decide how to create the
Outside and Inside products to add to
give the 15.
Factor: 2x 2  15x  7
1. List all of the pairs of factors of 2 and the pairs of
factors of 7. Note that each number has only one
pair of factors.
First
1, 2 because (1)(2) = 2
Last
1, 7 because (1)(7) = 7
2. Place the 1 and 2 with the x in the First position
in the two binomials.
(1x
) ( 2x
)
F L F L
3. Now decide how to place the 1 and 7 in Last positions
so that the Outside and Inside product add to give 15.
Factor: 2x 2  15x  7
4. There are two possibilities in which to place the
1 and 7 in the last (L) position in the two binomials..
(1x
1 ) ( 2x
7 ) or
( 1x
7 )( 2x
1)
5. Multiply the Outside and Inside terms and add them
together. Choose the combination that adds to 15.
2x
(1x
1 ) ( 2x
7x
7x + 2x = 9x
14x
7 ) or
( 1x
7 )( 2x
1)
1x
1x + 14x = 15x
Factor 2x 2+ 15x + 7
14x
2x
(1x
1 ) ( 2x
7x
( 1x
7)
or
7x + 2x = 9x
7 )( 2x
1)
1x
1x + 14x = 15x
6. The sign in the two binomials are the same and they
are both +, the sign of the middle coefficient.
( 1x + 7 )( 2x + 1 )
7. Check by FOILing: 2x2  1x  14x  7 = 2x 2 15x  7
Factor: 5x 2  38x  21
First
Last
1, 5
1, 21
3, 7
With one pair of factors from the First term and two pair of factors from the Last
term four possibilities for factoring exist.
Placing the possibilities in the two binomials and checking the sum of the
Outside and Inside products gives the following results.
L
L
L
F
F
F
L
F
( 1x 1 ) ( 5x 21 )
or
5x
L
(1x
F
3 ) ( 5x
15x
1)
105x
1x
21x
21x + 5x = 26x
F
( 1x 21 )( 5x
1x + 105x = 106x
F
L
F
L
L
7)
or
( 1x + 7 )( 5x + 3 )
7x
35x
3x
7x + 15x = 22x
3x + 35x = 38x
Factor:
5x 2  38 x + 21
Notice the pattern. It is possible to find the correct combination of Outside and
Inside products by working with only the numbers. Remember that green
represents Outside and blue represents Inside.
First
Last
1, 5
1, 21
21 + 5 = 26
1)
1, 5
1, 21
1 + 105 = 106
7)
1, 5
3, 7
7 + 15 = 22
7 ) (5x 3 )
1, 5
3, 7
3 + 35 = 38
5x
( 1x 1 ) ( 5x 21 )
21x
105x
(1x 21 )( 5x
1x
15x
( 1x
3 ) ( 5x
7x
35x
(1x
3x
35x
(1x
Factor: 5x 2  38x  21
Last
First
7 ) (5x 3 )
1, 5
3, 7 (1)(3) + (5)(7) = 38
3x
Begin by placing the First position in the two binomials.
) (5x
(1x
)
Then place the Last position in the two binomials so
that the Outside and Inside products add to give the
middle term in the trinomial.
35x
Finally place the signs
in the two binomials. (1x
+ 7 ) (5x + 3 )
3x
Check by FOILing:
5x 2  3x  35x  21 = 5x 2  38x  21
To factor the following trinomial consider the
following:
3x 2  13x  12
The − in this position
means that the signs in
the two binomial factors
will both be −.
The + in this position
means two things:
1.The sign in the two
binomials will be the
same.
2. You add the product
of the factors of 3 and 12
to get 13, the middle
coefficient.
Factor: 3x 2  13 x + 12
Notice that the Outside and Inside product are defined by the braces and not
by the position of the numbers in the First and Last columns.
First
Last
Outside
Inside
1, 3
1, 12
(1)(12)
(3)(1)
12 + 3 = 15
1, 3
1, 12
(1)(1)
(3)(12)
1 + 36 = 37
1, 3
2, 6
(1)(6)
(3)(2)
6 + 6 = 12
1, 3
2, 6
(1)(2)
(3)(6)
2 + 18 = 20
1, 3
3, 4
(1)(4)
(3)(3)
4 + 9 = 13
1, 3
3, 4
(1)(3)
(3)(4)
3 + 12 = 15
Sum
Factor: 3x 2  13 x + 12
First
1, 3
Outside + Inside
(1)(4) + (3)(3)
Last
3, 4
F
9x
Sum
4 + 9 = 13
F
(1x − 3 ) ( 3 x − 4 )
4x
Check by FOILing:
3x 2  9x  4x 12 = 3x 2 13x 12
TO REVIEW:
To factor a trinomial with leading coefficient not 1 with the
sign of the third term in the trinomial +:
ax 2  bx  c
or
ax 2  bx  c
1. List every pair of factors of the leading coefficient, a, and every pair
of factors of the third term, c.
2. Add the product of pairs of factors of a and c to determine which
combination add to give the middle coefficient, b.
3. The trinomial will be written as the product of two binomials with the
first (F) term being the factors of the leading coefficient, a, used in #2
combined with an x.
4. The last (L) terms in the two binomials will be the factors of c used
in #2. Once placed make sure that the Outside and Inside products
add to give b.
5. The sign in the two binomials will be the same as the sign of b.
To factor the following trinomial consider the
following:
5x 2  18x  8
The − in this position
means that the sign of
the largest of the Outside
or Inside product will be −.
The sign of the other
product will be +.
The − in this position
means two things:
1.The sign in the two
binomials will be different.
2. You subtract the product
of the factors of 5 and 8 to
get 18, the middle
coefficient.
5x 2  18 x − 8
Factor:
First
Last
Outside
Inside
Difference
1, 5
1, 8
(1)(8)
(5)(1)
8−5=3
1, 5
1, 8
(1)(1)
(5)(8)
40 − 1 = 39
1, 5
2, 4
(1)(4)
(5)(2)
10 − 4 = 6
1, 5
2, 4
(1)(2)
(5)(4)
20 − 2 = 18
Place the − in front
of the 4 to make the
larger product (20x)
negative.
Check by FOILing:
20x
(1x − 4 ) ( 5 x + 2 )
2x
5x 2  20x  2x  8 = 5x 2 18x  8
To factor the following trinomial consider the
following:
8x 2  3x  5
The + in this position
means that the sign of
the largest of the Outside
or Inside product will be +.
The sign of the other
product will be −.
The − in this position
means two things:
1.The sign in the two
binomials will be different.
2. You subtract the product
of the factors of 8 and 5 to
get 3, the middle coefficient.
8x 2  3 x − 5
Factor:
First
Last
Outside
Inside
Difference
1, 8
1, 5
(1)(5)
(8)(1)
8−5=3
1, 8
1, 5
(1)(1)
(8)(5)
40 − 1 = 39
2, 4
1, 5
(2)(5)
(4)(1)
10 − 4 = 6
2, 4
1, 5
(2)(1)
(4)(5)
20 − 2 = 18
Place the + in front
of the 1 to make the
larger product (8x)
positive.
8x
(1x + 1 ) ( 8 x − 5 )
5x
Check by FOILing: 8x 2  5x  8x  5 = 8x 2  3x  5
TO REVIEW:
To factor a trinomial with leading coefficient not 1 with the
sign of the third term in the trinomial −:
ax 2  bx  c
or
ax 2  bx  c
1. List every pair of factors of the leading coefficient, a, and every pair
of factors of the third term, c.
2. Subtract the product of pairs of factors of a and c to determine which
combination subtract to give the middle coefficient, b.
3. The trinomial will be written as the product of two binomials with the
first (F) term being the factors of the leading coefficient, a, used in #2
combined with an x.
4. The last (L) terms in the two binomials will be the factors of c used
in #2. Once placed make sure that the Outside and Inside products
subtract to give b.
5. The sign in the two binomials will be different and the sign of b is given
to the larger of the Outside or Inside product..
Factor:
6x 2  37x − 20
First
Last
Outside
Inside
Difference
1, 6
1, 6
1, 20
1, 20
(1)(20)
(1)(1)
(6)(1)
(6)(20)
20 − 6 = 14
120 − 1 = 119
1, 6
1, 6
2, 10
2, 10
(1)(10)
(1)(2)
(6)(2)
(6)(2)
12 − 10 = 2
12 − 2 = 10
1, 6
1, 6
4, 5
4, 5
(1)(5)
(1)(4)
(6)(4)
(6)(5)
24 − 5 = 19
30 − 4 = 26
2, 3
2, 3
1, 20
1, 20
(2)(20)
(2)(1)
(3)(1)
(3)(20)
40 − 3 = 37
60 − 2 = 58
2, 3
2, 3
2, 10
2, 10
(2)(10)
(2)(2)
(3)(2)
(3)(10)
20 − 6 = 14
30 − 4 = 26
2, 3
2, 3
4, 5
4, 5
(2)(5)
(2)(4)
(3)(4)
(3)(5)
12 − 10 = 2
15 − 8 = 7
6x 2  37x − 20
Factor:
First
2, 3
Last
Outside
Inside
Difference
(2)(20)
(3)(1)
40 − 3 = 37
1, 20
Place the − in front
of the 20 to make the
larger product (40x)
negative.
3x
(2x + 1 ) ( 3 x − 20 )
40x
Check by FOILing: 8x 2  40x  3x  20 = 8x 2  37x  20
10x 2  47xy + 21 y2
Factor:
First
Last
Outside
Inside
Sum
1, 10
1, 21
(1)(21)
(10)(1)
21 + 10 = 31
1, 10
1, 21
(1)(1)
(10)(21)
210 + 1 = 211
1, 10
3, 7
(1)(7)
(10)(3)
30 + 7 = 37
1, 10
3, 7
(1)(3)
(10)(7)
70 + 3 = 73
2, 5
1, 21
(2)(21)
(5)(1)
42 + 5 = 47
2, 5
1, 21
(2)(1)
(5)(21)
105 + 2 = 107
2, 5
3, 7
(2)(7)
(5)(3)
15 + 14 = 29
2, 5
3, 7
(2)(3)
(5)(7)
35 + 6 = 41
5xy
Both signs are the
same and both +.
(2 x + 1y ) ( 5 x + 21y )
42xy
Check by FOILing:
10x2  42xy 5xy  21y2 = 10x2  47xy  21y2
Factor: 28x 4 y  64x3y  60x 2 y
2
The Greatest Common Factor is 4 x y. Factoring gives the following:
4x 2 y(7x 2  16 x  15)
The next step is to factor the remaining trinomial.
First
Last
1, 7
1, 15
(1)(15)
(7)(1)
15 − 7 = 8
1, 7
1, 15
(1)(1)
(7)(15)
105 − 1 = 104
1, 7
3, 5
(1)(5)
(7)(3)
21 − 5 = 16
1, 7
3, 5
(1)(3)
(7)(5)
35 − 3 = 32
Outside
Inside
Difference
21x
Place the + in front of the 1 to
make the larger product (21xy)
positive.
4x 2 y (1 x + 3 ) ( 7 x − 5 )
5x
Check by FOILing and distributing the 4 x 2 y :
4 x 2 y(7 x 2  5x  21x  15) = 4 x 2 y(7 x 2  16x  15) = 28x 4 y  64x 3y  60x 2 y
FACTOR THE FOLLOWING POLYNOMIALS COMPLETELY:
To check your answer click on the arrow beside the problem.
1. 5x2  2x  7
2. 7x2  32x  36
3. 18x3  42x 2  24x
4. 6a 2  7ab  3b2
To go to the end of the tutorial click here:
1. (5x + 7)(x − 1)
To see the solution click here:
To go back to the problem page click here:
1. 5x2  2x  7
First
Last
Outside
Inside
Difference
1, 5
1, 7
(1)(7)
(5)(1)
7−5=2
1, 5
1, 7
(1)(1)
(5)(7)
35 − 1 = 34
( 1x − 1 ) ( 5 x + 7 )
5x
7x
To go back to problem page click here:
2. (7x − 18)(x − 2)
To see the solution click here:
To go back to the problem page click here:
2. 7x2  32x  36
First
Last
1, 7
1, 36
(1)(36)
(7)(1)
36 + 7 = 42
1, 7
1, 36
(1)(1)
(7)(36)
1 + 252 = 253
18 + 14 = 32
Outside
Inside
Sum
1, 7
2, 18
(1)(18)
(7)(2)
1, 7
2, 18
(1)(2)
(7)(18)
2 + 126 = 128
1, 7
3, 6
(1)(6)
(7)(3)
6 + 21 = 27
1, 7
3, 6
(1)(3)
(7)(6)
3 + 42 = 45
1, 7
4, 9
(1)(9)
(7)(4)
9 + 28 = 37
1, 7
4, 9
(1)(4)
(7)(9)
4 + 63 = 67
( 1 x − 2 ) ( 7 x − 18 )
14x
18x
To go back to the problem page click here:
3. 6x(x + 1)(3x + 4)
To see the solution click here:
To go back to the problem page click here:
3. 18x3 42x2  24x
The Greatest Common Factor is 6x. Factoring gives the following:
6x(3x 2  7 x 4)
First
Last
Outside
Inside
Sum
1, 3
1, 4
(1)(4)
(3)(1)
4+ 3 = 7
1, 3
1, 4
(1)(1)
(3)(4)
1+ 12= 13
1, 3
2, 2
(1)(2)
(3)(2)
2+ 6 = 8
3x
2x ( 1 x + 1 ) ( 3 x + 4 )
4x
To go back to the problem page click here:
4. (2a − 3b)(3a + b)
To see the solution click here:
To go back to the problem page click here:
4. 6a 2  7ab  3b2
First
Last
1, 6
1, 3
(1)(3)
(6)(1)
6−3=3
1, 6
1, 3
(1)(1)
(6)(3)
18 − 1 = 17
2, 3
1, 3
(2)(3)
(3)(1)
6−3=3
2, 3
1, 3
(2)(1)
(3)(3)
9−2=7
Outside
Inside
Difference
9ab
Place the – in front of the 3 to
make the larger product (9ab)
positive.
(2 a – 3 b) ( 3 a + 1 b )
2ab
To go back to the problem page click here:
See the instructor to obtain the Post-Quiz