Factoring notes

1. Factoring

2. Factoring

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

18. Solution #1: t2 – 4t – 21
1) Factors of -21: 1 • -21, -1 • 21
3 • -7, -3 • 7
2) Which pair adds to (- 4)?
3) Write the factors.
t2 – 4t – 21 = (t + 3)(t - 7)

19. Solution #2: x2 + 12x + 32
1) Factors of 32: 1 • 32
2 • 16
4•8
2) Which pair adds to 12 ?
3) Write the factors.
x2 + 12x + 32 = (x + 4)(x + 8)

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)

21. Solution #4: x2 + 3x - 18
1) Factors of -18: 1 • -18, -1 • 18
2 • -9, -2 • 9
3 • -6, -3 • 6
2) Which pair adds to 3 ?
3) Write the factors.
x2 + 3x - 18 = (x - 3)(x + 18)

22. Solution #5: 2x2 + x - 21
1) Multiply 2 • (-21) = - 42; 1 • -42, -1 • 42
list factors of - 42. 2 • -21, -2 • 21
3 • -14, -3 • 14
2) Which pair adds to 1 ? 6 • -7, -6 • 7
3) Write the temporary factors. ( x - 6)( x + 7)
2 2
4) Put “2” underneath. 3
( x - 6)( x + 7)
5) Reduce (if possible).
2 2
6) Move denominator(s)in ( x - 3)( 2x + 7)
front of “x”.
2x2 + x - 21 = (x - 3)(2x + 7)

23. Solution #6: 3x2 + 11x + 10
1) Multiply 3 • 10 = 30; 1 • 30
list factors of 30. 2 • 15
3 • 10
2) Which pair adds to 11 ? 5•6
3) Write the temporary factors. ( x + 5)( x + 6)
3 3
4) Put “3” underneath. 2
( x + 5)( x + 6)
5) Reduce (if possible).
3 3
6) Move denominator(s)in ( 3x + 5)( x + 2)
front of “x”.
3x2 + 11x + 10 = (3x + 5)(x + 2)