Factoring Trinomials
Multiply. (x+3)(x+2)
x2 + 2x + 3x + 6
Multiplying Binomials Use Foil
x2+ 5x + 6
Distribute.
x + 3
x
+
2
Using Algebra Tiles, we have:
= x2 + 5x + 6
Multiplying Binomials (Tiles)
Multiply. (x+3)(x+2)
x2 x
x 1
x x
x
1 1
1 1 1
How can we factor trinomials such as x2 + 7x + 12 back into binomials?
One method is algebra tiles: .
1) Start with x2.
Factoring Trinomials (Tiles)
2) Add seven “x” tiles (vertical or horizontal, at
least one of each) and twelve “1” tiles.
x2 x x xxx
x
x
1 1 1
1 1 1
1 1
1 1
1 1Rearrange until it is a rectangle.
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.
Factoring Trinomials (Tiles)
2) Add seven “x” tiles (vertical or horizontal, at
least one of each) and twelve “1” tiles.
x2 x x xxx
x
x
1 1 1
1 1 1
1 1
1 1
1 13) Rearrange the tiles until they form a rectangle!
We need to change the “x” tiles so the “1” tiles will fill in a rectangle.
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.
Factoring Trinomials (Tiles)
2) Add seven “x” tiles (vertical or horizontal, at
least one of each) and twelve “1” tiles.
x2 x x xxx
x 1 1 1
1 1 1
1 1
1 1 1
1
3) Rearrange the tiles until they form a rectangle!
Still not a rectangle.
x
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.
Factoring Trinomials (Tiles)
2) Add seven “x” tiles (vertical or horizontal, at
least one of each) and twelve “1” tiles.
x2 x x xx
x 1 1 1
1 1 1
1
1
1
1 113) Rearrange the tiles until they form a rectangle! A rectangle!!!
x
x
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’sThe # of “x” and “1” columns = constant.
Factoring Trinomials (Tiles)
5) Side factor:The # of x2 tiles = x’sThe # of “x” and “1” rows = constant.
x2 x x xx
x 1 1 1
1 1 1
1
1
1
1 11
x2 + 7x + 12 = ( x + 4)( x + 3)
x
x
x + 4
x
+
3
Again, we will factor trinomials such as x2 + 7x + 12 back into binomials.
look for the pattern of products and sums!
Factoring Trinomials
If the x2 term has no coefficient (other than 1)...
Step 1: What multiplies to the last term: 12?
x2 + 7x + 12
12 = 1 • 12
= 2 • 6
= 3 • 4
Factoring Trinomials
Step 2: The third term is positive so it must add to the middle term: 7?
x2 + 7x + 12
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 + )( x + )3 4
x2 + 7x + 12 = ( x + 3)( x + 4)
Factor. x2 + 2x - 24
This time, the last term is negative!
Factoring Trinomials
Step 1: Multiplies to 24. 24 = 1 • 24,
= 2 • 12,
= 3 • 8,
= 4 • 6,
4 – 6 = -2 6 – 4 = 2
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.
x2 + 2x - 24 = ( x - 4)( x + 6)
Factor. 3x2 + 14x + 8
This time, the x2 term has a coefficient (other than 1)!
Factoring Trinomials
Step 2: List all numbers that multiply to 24.
24 = 1 • 24
= 2 • 12
= 3 • 8
= 4 • 6
Step 4: Which pair adds up to 14?
Step 1: Multiply 3 • 8 = 24 (the leading coefficient & constant).
Step 3: When the last term is positive the signs are the same.
( 3x + 2 )( x + 4 )
2
Factor. 3x2 + 14x + 8continued
Factoring Trinomials
Step 5: Put the original leading coefficient (3) under both numbers.
( x + )( x + )
Step 6: Reduce the fractions, if possible.
Step 7: Move denominators in front of x.
Step 4: Write the factors. Both signs are positive.
123 3
2( x + )( x + )123 3
4
2( x + )( x + )43
( 3x + 2 )( x + 4 )
Factor. 3x2 + 14x + 8continued
Factoring Trinomials
You should always check the factors by distributing, especially since this process has more than a couple of steps.
= 3x2 + 14 x + 8
= 3x2 + 12x 2x + 8
√
3x2 + 14x + 8 = (3x + 2)(x + 4)
Factor 3x2 + 11x + 4
x2 has a coefficient (other than 1)!
Factoring Trinomials
Step 2: List all the factors of 12.
12 = 1 • 12
= 2 • 6
= 3 • 4
Step 3: Which pair adds up to 11? NoneIf it was 13x, 8x, or 7x, then it could be factored.
Step 1: Multiply 3 • 4 = 12 (the leading coefficient & constant).
Because None of the pairs add up to 11, this trinomial can’t be factored; it is PRIME.
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
POP QUIZ!
Solution #1: t2 – 4t – 21
1) Factors of 21: 1 • 213 • 7 3 - 7 or 7 - 3
2) Which pair subtracts to - 4?
3) Signs are mixed.
t2 – 4t – 21 = (t + 3)(t - 7)
Solution #2: x2 + 12x + 32
1) Factors of 32: 1 • 322 • 164 • 8
2) Which pair adds to 12 ?
3) Write the factors.
x2 + 12x + 32 = (x + 4)(x + 8)
Solution #3: x2 - 10x + 24
1) Factors of 32: 1 • 242 • 123 • 84 • 6
2) Both signs negative and adds to 10 ?
3) Write the factors.
x2 - 10x + 24 = (x - 4)(x - 6)
-1 • -24-2 • -12-3 • -8-4 • -6
Solution #4: x2 + 3x - 18
1) Factors of 18 and subtracts to 3.
1 • 18 2 • 9 3 • 6
2) The last term is negative so the signs are mixed.
3) Write the factors.
x2 + 3x - 18 = (x - 3)(x + 6)
3 – 6 = - 3
-3 + 6 = 3
Solution #5: 2x2 + x - 21
1) factors of 42. 1 • 42 2 • 213 • 14 6 • 76 – 7 = -17 – 6 = 1 2) subtracts to 1
3) Signs are mixed.
2x2 + x - 21 = (x - 3)(2x + 7)
( x - 6)( x + 7)
4) Put “2” underneath.2 2
5) Reduce (if possible). ( x - 6)( x + 7)2 2
3
6) Move denominator(s)to the front of “x”.
( x - 3)( 2x + 7)
Solution #6: 3x2 + 11x + 10
1) Multiply 3 • 10 = 30; list factors of 30.
1 • 302 • 153 • 105 • 62) Which pair adds to 11 ?
3) The signs are both positive
3x2 + 11x + 10 = (3x + 5)(x + 2)
( x + 5)( x + 6)
4) Put “3” underneath.3 3
5) Reduce (if possible). ( x + 5)( x + 6)3 3
2
6) Move denominator(s)in front of “x”.
( 3x + 5)( x + 2)
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