Download x 2

Factoring Trinomials
Multiplying Binomials
Use Foil
Multiply. (x+3)(x+2)
Distribute.
x2 + 2x
+ 3x + 6
x2+ 5x + 6
Multiplying Binomials (Tiles)
Multiply. (x+3)(x+2)
Using Algebra Tiles, we have:
x +
x
x2
3
x
x
x
= x2 + 5x + 6
+
x
1
1
1
2
x
1
1
1
Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is algebra tiles: .
1) Start with x2.
x2
x
x
x
x
x
(vertical or horizontal, at
least one of each) and
x
1
1
1 1
1
twelve “1” tiles.
x
1
1
1
1
1
1
2) Add seven “x” tiles
Rearrange until it is a rectangle.
1
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
(vertical or horizontal, at
least one of each) and
x
1
1
1 1
1
twelve “1” tiles.
x
1
1
1
1
1
1
2) Add seven “x” tiles
3) Rearrange the tiles
until they form a
rectangle!
1
We need to change the “x” tiles so
the “1” tiles will fill in a rectangle.
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.
2) Add seven “x” tiles
(vertical or horizontal, at
least one of each) and
twelve “1” tiles.
3) Rearrange the tiles
until they form a
rectangle!
x2
x
x
x
x
x
x
x
1
1
1 1
1
1
1
1
1
1
1
1
Still not a rectangle.
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
(vertical or horizontal, at
least one of each) and
x
1
1
1 1
twelve “1” tiles.
x
1
1
1
1
x
1
1
1
1
2) Add seven “x” tiles
3) Rearrange the tiles
until they form a
rectangle!
A rectangle!!!
x
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:
The # of x2 tiles = x’s
The # of “x” and “1”
columns = constant.
5) Side factor:
The # of x2 tiles = x’s
The # of “x” and “1”
rows = constant.
x
+ 4
x
x2
x
x
x
+
x
1
1
1 1
3
x
1
1
1
1
x
1
1
1
1
x
x2 + 7x + 12 = ( x + 4)( x + 3)
Factoring Trinomials
Again, we will factor trinomials such as
x2 + 7x + 12 back into binomials.
look for the pattern of products and sums!
If the x2 term has no coefficient (other than 1)...
x2 + 7x + 12
Step 1:
What multiplies to the last term: 12?
12 = 1 • 12
=2•6
=3•4
Factoring Trinomials
x2 + 7x + 12
Step 2: The third term is
positive so it must add to the
middle term: 7?
12 = 1 • 12
=2•6
=3•4
Step 3: The third term is
positive so the signs are both
the same as the middle term.
Both positive.
( x + 3 )( x + 4 )
x2 + 7x + 12 = ( x + 3)( x + 4)
Factoring Trinomials
Factor. x2 + 2x - 24
This time, the last term is negative!
Step 1: Multiplies to 24.
24 = 1 • 24,
Step 2: The third term is
negative. That means it subtracts
to the middle number and has
mixed signs.
Step 3: Write the binomial
factors and then check
your answer.
= 2 • 12,
= 3 • 8,
= 4 • 6,
4 – 6 = -2
6–4=2
x2 + 2x - 24 = ( x - 4)( x + 6)
Factoring Trinomials
Factor.
3x2 + 14x + 8
This time, the x2 term has a coefficient (other than 1)!
Step 1: Multiply 3 • 8 = 24
(the leading coefficient & constant).
24 = 1 • 24
= 2 • 12
Step 2: List all numbers that
multiply to 24.
Step 3: When the last term is
positive the signs are the same.
Step 4: Which pair adds up to 14?
=3•8
=4•6
Factoring Trinomials
3x2 + 14x + 8
Factor.
continued
Step 4: Write the factors.
Both signs are positive.
( 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
Factor.
3x2 + 14x + 8
continued
You should always check the factors by distributing, especially
since this process has more than a couple of steps.
( 3x + 2 )( x + 4 ) = 3x2 + 12x
2x + 8
= 3x2 + 14 x + 8
√
3x2 + 14x + 8 = (3x + 2)(x + 4)
Factoring Trinomials
Factor 3x2 + 11x + 4
x2 has a coefficient (other than 1)!
Step 1: Multiply 3 • 4 = 12
(the leading coefficient & constant).
Step 2: List all the factors of 12.
12 = 1 • 12
=2•6
=3•4
Step 3: Which pair adds up to 11? None
If it was 13x, 8x, or 7x, then it could be factored.
Because None of the pairs add up to 11,
this trinomial can’t be factored; it is PRIME.
POP QUIZ!
Factor these trinomials: watch your 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
Solution #1:
t2 – 4t – 21
1 • 21
3 • 7 3 - 7 or 7 - 3
1) Factors of 21:
2) Which pair subtracts to - 4?
3) Signs are mixed.
t2 – 4t – 21 = (t + 3)(t - 7)
Solution #2:
x2 + 12x + 32
1 • 32
2 • 16
4•8
1) Factors of 32:
2) Which pair adds to 12 ?
3) Write the factors.
x2 + 12x + 32 = (x + 4)(x + 8)
Solution #3:
1 • 24
2 • 12
3•8
4•6
1) Factors of 32:
x2 - 10x + 24
-1 • -24
-2 • -12
-3 • -8
-4 • -6
2) Both signs negative and
adds to 10 ?
3) Write the factors.
x2 - 10x + 24 = (x - 4)(x - 6)
Solution #4:
1) Factors of 18
and subtracts to 3.
x2 + 3x - 18
1 • 18
2•9
3•6
2) The last term is negative so
the signs are mixed.
3–6=-3
3) Write the factors.
-3 + 6 = 3
x2 + 3x - 18 = (x - 3)(x + 6)
Solution #5:
1) factors of 42.
2) subtracts to 1
3) Signs are mixed.
4) Put “2” underneath.
2x2 + x - 21
1 • 42
2 • 21
3 • 14
6•7
6 – 7 = -1
7–6=1
( x - 6)( x + 7)
2
2
3
5) Reduce (if possible).
( x - 6)( x + 7)
2
2
6) Move denominator(s)
to the front of “x”.
( x - 3)( 2x + 7)
2x2 + x - 21 = (x - 3)(2x + 7)
Solution #6:
1) Multiply 3 • 10 = 30;
list factors of 30.
2) Which pair adds to 11 ?
3) The signs are both positive
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)