9-5 Factoring x2 + bx + c

Factoring is the inverse of multiplying. We are rewriting a polynomial as the product of 2

factors.

Definition

Remember when we multiplied, the “c” was what the two factors (or last terms) MULTIPLIED to.

The “b” was the “OI” of the FOIL process. This is what was ADDED together.So, we are looking for two numbers that when we multiply we get “c”, but when

we add, we get “b”. Those will become our factors!

Factoring x2 + bx + c

Example #1Factor: x2 + 7x + 12

We are looking for two numbers that when we multiply we get 12, but when we

add, we get 7. What are all the ways of getting 12?1·122·63·4

Factoring x2 + bx + c

Which pair adds to 7?Finally, write the factors

(x+3)(x+4)

Example #2Factor: y2 + 6y – 27

We are looking for two numbers that multiply to -27, but add to 6.

What are all the ways of getting -27?

-1·27-3·93·-91·-27

Factoring x2 + bx + c

Which pair adds to 6?Finally, write the factors

(y-3)(y+9)

Example #3Factor: p2 – 2p – 15

We are looking for two numbers that multiply to -15, but add to -2.

What are all the ways of getting -15?

-1·15-3·53·-51·-15

Factoring x2 + bx + c

Which pair adds to -2?Finally, write the factors

(p+3)(p-5)

Example #4Factor: p2 – 2rp – 15r2

We are looking for two numbers that multiply to -15, but add to -2.

What are all the ways of getting -15?

-1·15-3·53·-51·-15

Factoring x2 + bx + c

Which pair adds to -2?Finally, write the factors

(p+3r)(p-5r)

Example #5Factor: k2 – 13k + 12

We are looking for two numbers that multiply to 12, but add to -13.

What are all the ways of getting 12?1·122·63·4but none add to -13

Factoring x2 + bx + c

Which pair adds to -13 if both are negatives?Finally, write the factors

(k-1)(k-12)

Today’s Assignment

Box Method for Factoring x2 + bx + c

Enter 1st term and last term in the diagonal top left to bottom right.

1st term

last term

Box Method for Factoring x2 + bx + c

Look at c, the last term (this is what the factors must multiply to)

1st term

last term

Box Method for Factoring x2 + bx + c

b is what the factors must add to

1st term

last term

Box Method for Factoring x2 + bx + c

So we look for 2 numbers that multiply to get c and add to get b and enter them into the other diagonals (don’t forget to include the variable.)

1st term

last termfactor

factor

Box Method for Factoring x2 + bx + c

Finally, we find the GCF of each row and column…those become the factors of x2 + bx + c.

1st term

last termfactor

factorGCF

GCF GCF

GCF

Example #5

Factor: x2 + 8x + 7

Box Method for Factoring x2 + 8x + 7

Enter 1st term and last term in the diagonal top left to bottom right.

x2

7

Box Method for Factoring x2 + 8x + 7

Find c (this is what the factors must multiply to)

x2

7

c = 7

Box Method for Factoring x2 + 8x + 7

b is what the factors must add to

x2

7

c = 7

Box Method for Factoring x2 + 8x + 7

So we look for 2 numbers that multiply to get 7 and add to get 8 and enter them into the other diagonals (don’t forget to include the variable.)

x2

7

c = 7

1x

7x

Box Method for Factoring x2 + 8x + 7

Finally, we find the GCF of each row and column…those become the factors of x2 + 8x + 7.

x2

7

c = 7

1x

7xx

1

x 7

(x+1)(x+7)