3. 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
4. Multiplying Binomials (Tiles)
Multiply. (x+3)(x+2)
Using Algebra Tiles, we have:
x + 3
x x2 x x x
+ = x2 + 5x + 6
x 1 1 1
2 x 1 1 1
5. Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is to again use algebra tiles:
1) Start with x2.
x2 x x x x x
2) Add seven “x” tiles
(vertical or horizontal, at
x 1 1 1 1 1
least one of each) and
twelve “1” tiles. x 1 1 1 1 1
1 1
6. Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is to again use algebra tiles:
1) Start with x2.
x2 x x x x x
2) Add seven “x” tiles
(vertical or horizontal, at
x 1 1 1 1 1
least one of each) and
twelve “1” tiles. x 1 1 1 1 1
3) Rearrange the tiles 1 1
until they form a
We need to change the “x” tiles so
rectangle!
the “1” tiles will fill in a rectangle.
7. Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is to again use algebra tiles:
1) Start with x2.
x2 x x x x x x
2) Add seven “x” tiles
(vertical or horizontal, at
x 1 1 1 1 1 1
least one of each) and
twelve “1” tiles. 1 1 1 1 1 1
3) Rearrange the tiles
until they form a Still not a rectangle.
rectangle!
8. Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is to again use algebra tiles:
1) Start with x2.
x2 x x x x
2) Add seven “x” tiles
(vertical or horizontal, at
x 1 1 1 1
least one of each) and
twelve “1” tiles. x 1 1 1 1
x 1 1 1 1
3) Rearrange the tiles
until they form a
rectangle! A rectangle!!!
9. Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is to again use algebra tiles:
4) Top factor: x + 4
The # of x2 tiles = x’s x x2 x x x x
The # of “x” and “1”
columns = constant. + x 1 1 1 1
3 x 1 1 1 1
5) Side factor:
The # of x2 tiles = x’s x 1 1 1 1
The # of “x” and “1”
rows = constant.
x2 + 7x + 12 = ( x + 4)( x + 3)
10. Factoring Trinomials (Method 2)
Again, we will factor trinomials such as
x2 + 7x + 12 back into binomials.
This method does not use tiles, instead we look
for the pattern of products and sums!
If the x2 term has no coefficient (other than 1)...
x2 + 7x + 12
Step 1: List all pairs of 12 = 1 • 12
numbers that multiply to
=2•6
equal the constant, 12.
=3•4
11. Factoring Trinomials (Method 2)
x2 + 7x + 12
Step 2: Choose the pair that 12 = 1 • 12
adds up to the middle
=2•6
coefficient.
=3•4
Step 3: Fill those numbers
into the blanks in the ( x + 3 )( x + 4 )
binomials:
x2 + 7x + 12 = ( x + 3)( x + 4)
12. Factoring Trinomials (Method 2)
Factor. x2 + 2x - 24
This time, the constant is negative!
Step 1: List all pairs of -24 = 1 • -24, -1 • 24
numbers that multiply to equal
the constant, -24. (To get -24, = 2 • -12, -2 • 12
one number must be positive and = 3 • -8, -3 • 8
one negative.)
= 4 • -6, - 4 • 6
Step 2: Which pair adds up to 2?
Step 3: Write the binomial x2 + 2x - 24 = ( x - 4)( x + 6)
factors.
13. Factoring Trinomials (Method 2*)
Factor. 3x2 + 14x + 8
This time, the x2 term DOES have a coefficient (other than 1)!
Step 1: Multiply 3 • 8 = 24 24 = 1 • 24
(the leading coefficient & constant).
= 2 • 12
Step 2: List all pairs of =3•8
numbers that multiply to equal
that product, 24. =4•6
Step 3: Which pair adds up to 14?
14. Factoring Trinomials (Method 2*)
Factor. 3x2 + 14x + 8
Step 4: Write temporary ( x + 2 )( x + 12 )
factors with the two numbers. 3 3
Step 5: Put the original 4
leading coefficient (3) under ( x + 2 )( x + 12 )
both numbers. 3 3
Step 6: Reduce the fractions, if ( x + 2 )( x + 4 )
possible. 3
Step 7: Move denominators in ( 3x + 2 )( x + 4 )
front of x.
15. 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)
16. 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 12 = 1 • 12
(the leading coefficient & constant).
=2•6
Step 2: List all pairs of
numbers that multiply to equal =3•4
that product, 12.
Step 3: Which pair adds up to 11?
None of the pairs add up to 11, this trinomial
can’t be factored; it is PRIME.
17. 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) t2 – 4t – 21
2) x2 + 12x + 32
3) x2 –10x + 24
4) x2 + 3x – 18
5) 2x2 + x – 21
6) 3x2 + 11x + 10
20. Solution #3: x2 - 10x + 24
1) Factors of 32: 1 • 24 -1 • -24
2 • 12 -2 • -12
3•8 -3 • -8
4•6 -4 • -6
2) Which pair adds to -10 ? None of them adds to (-10). For
the numbers to multiply to +24
and add to -10, they must both be
negative!
3) Write the factors.
x2 - 10x + 24 = (x - 4)(x - 6)
