Section 6.2
The more familiar we are with multiplication of
binomials, the easier factoring trinomials will be.
Recall:
To multiply 2 binomials, use the FOIL method:
(x + 3) (x + 4) = x2 + 7x + 12
(x – 5) (x + 2) = x2 – 3x – 10
Factoring Trinomials with a
Leading Coefficient of 1
To factor a trinomial in which the coefficient of x2
is 1, we will use the “Reverse FOIL” method.
We find two numbers a and b whose sum is the
coefficient of the middle term and whose product
is the constant term (last term).
Factor: x2 + 8x + 12.
We need two numbers whose sum is 8 and whose
product is 12.
The numbers are 6 and 2:
x2 + 8x + 12 = (x + 6) ( x + 2)
We can easily check our work by multiplying:
Check: (x + 6)(x + 2) = x2 + 6x + 2x + 12
= x2 + 8x + 12
Factoring Out the
Greatest Common Factor
Before Using
Reverse Foil
Factor: 2x2 + 10x – 28
The coefficient of x2 is 2.
We begin by factoring out the greatest common factor,
which is 2:
2x2 + 10x – 28 = 2 (x2 + 5x – 14)
Now, we factor the remaining trinomial by finding a
pair of numbers whose sum is 5 and whose product is
–14.
Here are the possibilities:
From the last line we see that the factors of x2 + 5x – 14
are (x + 7) and (x – 2).
The complete solution is:
2x2 + 10x – 28 = 2(x2 + 5x – 14)
= 2(x + 7)(x – 2)
The sign of the last term of the trinomial gives us information
about the binomial factors:
+ tells us the middle sign of both factors is the same
─ tells us the middle sign of both factors is different
a2 + 2ab + b2 = (
+
)(
+
)
a2 ─ 2ab + b2 = (
─
)(
─
)
a2 + 2ab ─ b2 = (
+
) (
─
)
The sign of the middle term in
the trinomial tells us whether
the middle signs of both
binomial factors are + or ─
The sign of the middle term in
the trinomial gives the sign of
the second term in the
binomial factors that has the
higher absolute value
Section 6.2
Pages 435-438
#1, 7, 9, 13, 21, 33, 37, 43, 47