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)