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Factoring Trinomials
Multiplying Binomials (FOIL)
Multiply. (x+3)(x+2)
Distribute.
x•x+x•2+3•x+3•2
F
O
I
L
= x2+ 2x + 3x + 6
= x2+ 5x + 6
Factoring Trinomials (Method 1*)
Factor. 3x2 + 14x + 8
This time, the x2 term DOES have a coefficient (other than 1)!
Step 1: Multiply 3 • 8 = 24
(the leading coefficient & constant).
24 = 1 • 24
= 2 • 12
Step 2: List all pairs of
numbers that multiply to equal
that product, 24.
Step 3: Which pair adds up to 14?
=3•8
=4•6
Factoring Trinomials (Method 1*)
Factor. 3x2 + 14x + 8
Step 4: Rewrite the middle
term with the two numbers.
Step 5: Group them into two
pairs
Step 6: Factor the greatest
common factor from each group
Step 7: Factor the common factor
from each term to form two
factors
( 3x + 2 )( x + 4 )
Factoring Trinomials (Method 2*)
Factor. 3x2 + 14x + 8
Step 1: Multiply 3 • 8 = 24
(the leading coefficient & constant).
24 = 1 • 24
= 2 • 12
Step 2: List all pairs of
numbers that multiply to equal
that product, 24.
Step 3: Which pair adds up to 14?
=3•8
=4•6
Factoring Trinomials (Method 2*)
Factor. 3x2 + 14x + 8
Step 4: Write temporary
factors with the two numbers.
( x + 2 )( x + 12 )
3
3
Step 5: Put the original
leading coefficient (3) under
both numbers.
( x + 2 )( x + 12 )
3
3
Step 6: Reduce the fractions, if
possible.
( x + 2 )( x + 4 )
3
Step 7: Move denominators in
front of x.
( 3x + 2 )( x + 4 )
4
Factoring Trinomials (Method 2*)
Factor.
3x2 + 14x + 8
You should always check the factors by distributing, especially
since this process has more than a couple of steps.
( 3x + 2 )( x + 4 ) = 3x • x + 3x • 4 + 2 • x + 2 • 4
= 3x2 + 14 x + 8
√
3x2 + 14x + 8 = (3x + 2)(x + 4)
Factoring Trinomials (Method 2*)
Factor 3x2 + 11x + 4
This time, the x2 term DOES have a coefficient (other than 1)!
Step 1: Multiply 3 • 4 = 12
(the leading coefficient & constant).
Step 2: List all pairs of
numbers that multiply to equal
that product, 12.
12 = 1 • 12
=2•6
=3•4
Step 3: Which pair adds up to 11?
None of the pairs add up to 11, this trinomial
can’t be factored over the rationals; it is PRIME.
Factor These Trinomials!
Factor each trinomial, if possible. The first four do NOT have
leading coefficients, the last two DO have leading coefficients.
Watch out for signs!!
1) 2x2 + x – 21
2) 3x2 + 11x + 10
Solution #1:
1) Multiply 2 • (-21) = - 42;
list factors of - 42.
2) Which pair adds to 1 ?
3) Write the temporary factors.
4) Put “2” underneath.
2x2 + x - 21
1 • -42, -1 • 42
2 • -21, -2 • 21
3 • -14, -3 • 14
6 • -7, -6 • 7
( x - 6)( x + 7)
2
2
3
5) Reduce (if possible).
( x - 6)( x + 7)
2
2
6) Move denominator(s)in
front of “x”.
( x - 3)( 2x + 7)
2x2 + x - 21 = (x - 3)(2x + 7)
Solution #2:
1) Multiply 3 • 10 = 30;
list factors of 30.
2) Which pair adds to 11 ?
3) Write the temporary factors.
4) Put “3” underneath.
3x2 + 11x + 10
1 • 30
2 • 15
3 • 10
5•6
( x + 5)( x + 6)
3
3
2
5) Reduce (if possible).
( x + 5)( x + 6)
3
3
6) Move denominator(s)in
front of “x”.
( 3x + 5)( x + 2)
3x2 + 11x + 10 = (3x + 5)(x + 2)