Section 14.2
FACTORING TRINOMIALS OF THE FORM
X 2+ B X + C
Factoring Trinomials of the Form x2 + bx + c
 Factor trinomials of the form x2 + bx + c
 Factor out the greatest common factor and then
factor a trinomial of the form x2 + bx + c
Factoring Trinomials of the Form x2 + bx + c
 Multiply the binomials
1. ( x  2)( x  5)
x 2  5 x  2 x  10
x 2  7 x  10
2. ( x  3)( x  4)
x 2  4 x  3x  12
x 2  x  12
 Trinomials of the form
x2 + bx + c result from
the product of two
binomials of the form
(x + m)(x + n).
Factoring Trinomials of the Form x2 + bx + c
 Multiply the binomials
1. ( x  2)( x  5)
x 2  5 x  2 x  10
x 2  7 x  10
2. ( x  3)( x  4)
x 2  4 x  3x  12
x 2  x  12
 Notice, the first term in
the trinomial comes from
the product of the first
terms in each binomial.
Factoring Trinomials of the Form x2 + bx + c
 Multiply the binomials
1. ( x  2)( x  5)
x 2  5 x  2 x  10
x 2  7 x  10
2. ( x  3)( x  4)
x 2  4 x  3x  12
x 2  x  12
 Notice, the first term in
the trinomial comes from
the product of the first
terms in each binomial.
 Notice, the last term in
the trinomial comes from
the product of the last
terms in each binomial.
Factoring Trinomials of the Form x2 + bx + c
 Multiply the binomials
1. ( x  2)( x  5)
x 2  5 x  2 x  10
x 2  7 x  10
2. ( x  3)( x  4)
x 2  4 x  3x  12
x 2  x  12
 Notice, the first term in
the trinomial comes from
the product of the first
terms in each binomial.
 Notice, the last term in
the trinomial comes from
the product of the last
terms in each binomial.
 Notice, the middle term
in the trinomial comes
from the sum of the last
terms in the binomial.
Factoring Trinomials of the Form x2 + bx + c
 Proof
 Multiply (x + m)(x + n)
 x  m x  n 
x 2  nx  mx  mn
x2   n  m x  mn
The middle term, the
coefficient of x, is the sum of
the constants in the
binomials.
The last term, the constant, is
the product of the constants in
the binomials.
Factoring Trinomials of the Form x2 + bx + c
 Trinomials of the form
 Factor
1.
x  19 x8  20
1 20
2
What
sums to
9?
27
3 6
...
1  10
41  50



x4 x5
2 10
Which of
45
these
What
1 20
multiplies
sum to 9?
2 10
to 20?
4  5
x2 + bx + c result from the
product of two binomials of the
form (x + m)(x + n) where
m+n=b and mn=c.
 To factor a trinomial:
1.
List all factor pairs of c

...
Limited
AAACK!
number of
Infinite
possibilities! options!
2.
3.


Start here since the amount
of factors for a number is
finite.
Find the pair whose sum or
difference is b
Check the factorization by
multiplying it back out
If c is positive, m and n will have the
same sign and will add to b
If c is negative, m and n will have
different signs and will subtract to b
Factoring Trinomials of the Form x2 + bx + c
 Trinomials of the form
 Factor
1.
x  9 x  20
2
 x  4 x  5
2.
x 2  13 x  22
 x  2 x 11
3.
x 2  5 x  36
 x  4 x  9
4.
x2 + bx + c result from the
product of two binomials of the
form (x + m)(x + n) where
m+n=b and mn=c.
 To factor a trinomial:
1.
List all factor pairs of c

2.
3.
x 2  3x  40
 x  8 x  5


Start here since the amount
of factors for a number is
finite.
Find the pair whose sum or
difference is b
Check the factorization by
multiplying it back out
If c is positive, m and n will have the
same sign and will add to b
If c is negative, m and n will have
different signs and will subtract to b
Factoring Trinomials of the Form x2 + bx + c
Special Cases
 Factor
1.
a 2  13ab  30b 2
 a 10b a  3b 
2.
y 2  6 y  15
prime

A polynomial that cannot be factored is called prime.
Factoring Out the Greatest Common Factor
 Factor
1.
x3  3x 2  4 x

2.
Always begin by factoring out the GCF if it exists.
x  x  1 x  4
5 x5  25 x 4  30 x3
5x3  x  6 x  1