Dividing Polynomials

Synthetic Division

Used to factor polynomials and find the roots / zeros.

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

1

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

-2

1

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

-2

1 -2

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

-2 4

1 -2

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

-2 4

1 -2 3

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

-2 4 -6

1 -2 3

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

-2 4 -6

1 -2 3 -5

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

-2 4 -6

1 -2 3 -5

x2 x constant

remainder

Synthetic Division

Same idea as synthetic substitution.

Remember to use 0 as the coefficient of any missing terms!

Divide f(x) = x3 – x + 1 by x + 2.

-2 1 0 -1 1

-2 4 -6

1 -2 3 -5

2x

532xx2

x2 x constant

remainder

Factoring

Use one factor to divide the polynomial and find the rest.

Factoring

Use one factor to divide the polynomial and find the rest.Completely factor f(x) = x3 – 10x2 + 19x + 30 given that f(6) = 0.

Factoring

Use one factor to divide the polynomial and find the rest.

First, divide f(x) = x3 – 10x2 + 19x + 30 by x – 6.

Completely factor f(x) = x3 – 10x2 + 19x + 30 given that f(6) = 0.

Factoring

Use one factor to divide the polynomial and find the rest.

First, divide f(x) = x3 – 10x2 + 19x + 30 by x – 6.

6 1 -10 19 30

6 -24 -30

1 -4 -5 0

Completely factor f(x) = x3 – 10x2 + 19x + 30 given that f(6) = 0.

Factoring

Use one factor to divide the polynomial and find the rest.

First, divide f(x) = x3 – 10x2 + 19x + 30 by x – 6.

6 1 -10 19 30

6 -24 -30

1 -4 -5 0x2 – 4x – 5

Completely factor f(x) = x3 – 10x2 + 19x + 30 given that f(6) = 0.

Factoring

Use one factor to divide the polynomial and find the rest.

First, divide f(x) = x3 – 10x2 + 19x + 30 by x – 6.

6 1 -10 19 30

6 -24 -30

1 -4 -5 0x2 – 4x – 5

Completely factor f(x) = x3 – 10x2 + 19x + 30 given that f(6) = 0.

Second, factor the binomial x2 – 4x – 5.

Factoring

Use one factor to divide the polynomial and find the rest.

First, divide f(x) = x3 – 10x2 + 19x + 30 by x – 6.

6 1 -10 19 30

6 -24 -30

1 -4 -5 0x2 – 4x – 5

Completely factor f(x) = x3 – 10x2 + 19x + 30 given that f(6) = 0.

Second, factor the binomial x2 – 4x – 5.

x2 – 4x – 5 = (x – 5)(x + 1)

Factoring

Use one factor to divide the polynomial and find the rest.

First, divide f(x) = x3 – 10x2 + 19x + 30 by x – 6.

6 1 -10 19 30

6 -24 -30

1 -4 -5 0x2 – 4x – 5

Completely factor f(x) = x3 – 10x2 + 19x + 30 given that f(6) = 0.

Second, factor the binomial x2 – 4x – 5.

x2 – 4x – 5 = (x – 5)(x + 1)

So f(x) = (x – 6)(x – 5)(x + 1).

Practice

Use synthetic division to completely factor

f(x) = x3 – 10x2 + 8x + 64 given that f(-2) = 0.