7.4 the remainder and factor theorems

7.4 The Remainder and Factor Theorems

Use Synthetic Substitution to find Remainders

A review of Synthetic Substitution

Synthetic Division a method of dividing polynomials using the coefficients of the divided polynomial and its divisor.

Synthetic Division definition

Synthetic Division a method of dividing polynomials using the coefficients of the divided polynomial and its divisor.

Since the divided polynomial is in descending order, we can just use the coefficient as they are written.

4642 23 xxxx

Synthetic Division definition

2 | 1 -4 6 -4 Inside the box is the zero of x - 2

Synthetic can only be used when the divisor has a degree of 1

Synthetic Division definition

2 | 1 -4 6 -4 Inside the box is 2 -4 4 the zero of x - 2

1 - 2 2 0

We drop the first number the then multiply by 2 and add it to the next number. Repeat till done.

Synthetic Division definition

2 | 1 -4 6 -4 Inside the box is 2 -4 4 the zero of x - 2

1 - 2 2 0

We drop the first number the then multiply by 2 and add it to the next number. Repeat till done.

Synthetic Division definition

2 | 1 -4 6 -4 Inside the box is 2 -4 4 the zero of x - 2

1 - 2 2 0

Since the Remainder is Zero, (x – 2) is a factor of 464 23 xxx

A new way to Evaluate polynomials

Find

Use Synthetic Substitution

223)4( 234 xxxf

A new way to Evaluate polynomials

Find

Use Synthetic Substitution

4 | 3 -2 1 0 - 2

12

3

223)4( 234 xxxf

A new way to Evaluate polynomials

Find

Use Synthetic Substitution

4 | 3 -2 1 0 - 2

12 40

3 10

223)4( 234 xxxf

A new way to Evaluate polynomials

Find

Use Synthetic Substitution

4 | 3 -2 1 0 - 2

12 40 164

3 10 41

223)4( 234 xxxf

A new way to Evaluate polynomials

Find

Use Synthetic Substitution

4 | 3 -2 1 0 - 2

12 40 164 656

3 10 41 164

223)4( 234 xxxf

A new way to Evaluate polynomials

Find

Use Synthetic Substitution

4 | 3 -2 1 0 - 2

12 40 164 656

3 10 41 164 654

223)4( 234 xxxf

A new way to Evaluate polynomials

Find

Use Synthetic Substitution

4 | 3 -2 1 0 - 2

12 40 164 656

3 10 41 164 654

Remainder is 654, so f(4) = 654

223)4( 234 xxxf

To show a binomial is a Factor the remainder equals zero

Show (x – 3) is a factor of

x3 + 4x2 – 15x – 18.

To show a binomial is a Factor the remainder equals zero

Show (x – 3) is a factor of

x3 + 4x2 – 15x – 18.

3 | 1 4 -15 -18

3 21 18

1 7 6 0

Since the remainder is zero, ( x – 3) is a factor

To show a binomial is a Factor the remainder equals zero

Show (x – 3) is a factor of

x3 + 4x2 – 15x – 18.

3 | 1 4 -15 -18

3 21 18

1 7 6 0

If you were going to find the other factors, you would use the depressed equation.

x2 + 7x + 6 = (x + 6)(x + 1)

equationdepressed

x3 + 4x2 – 15x – 18 factor

Using the synthetic substitution and factoring the depressed equation gives us the factors of the polynomial.

x3 + 4x2 – 15x – 18 = (x – 3)(x + 1)(x + 6)

Homework

Page 368 – 369

# 13 – 29 odd

Homework

Page 368 – 369

# 14 – 30 even