October 26, 2017

3.2 - Synthetic Division and Remainder Theorems

Example 1: Use long division to divide 875 by 3

Key Terms:

Divisor

Dividend

Quotient

Remainder

Dividend

Divisor

Quotient Remainder

This means the answer is written as such:

Divisor

Dividend

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Division can also be written as:Dividend = (divisor)(quotient)+remainder

What if we wanted to undo the division? What would we multiply both sides by?

If the remainder is 0, then the divisor is a factor of the dividend

October 26, 2017

Example 2: Divide 5x2-4x+17 by x-1. Write your answer as:

a)

b)

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Example 3: Divide 2x3+4x+7 by x2-1. Write your answer as:

a)Remember to use place-holder zeros for missing values!

b)

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Synthetic Division-Can only be used if the divisor is a linear factor (x+b). Divide 5x2-4x+17 by x-1

1 5 4 17zero of th

e divisor coefficients of

the dividend

5

Example 4: Use synthetic division to divide -4x3+2x2-11 by x-3

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b) Is x+1 a factor of P(x)?

Example 5:

a) Find the remainder if p(x)= 2x3+4x2-8x-15 is divided by (x+1) using division

c) Evaluate p(-1). What do you notice?

October 26, 2017

The Remainder TheoremIf a polynomial p(x) is divided by (x-c), then the remainder is equal to p(c).

Example 6: Use the remainder theorem to evaluate p(x)=2x4-5x3+2x2+x-8 for p(-5). Verify with substitution.

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Example 7: Use the remainder theorem to evaluate the following: f(x)=x4+x3-10x2-4x+24

a) f(-1)

b) f(2)

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What does a remainder of 0 mean????

*** If the remainder is 0, then (x-c) is a factor of p(x) *** This is called the Factor Theorem

The Factor Theorem:

1. If p(c)=0, then x-c is a factor of p(x)

2. If x-c is a factor of p(x), then p(c)=0.

October 26, 2017

Example 8: Determine if the following are factors of g(x)=x3-13x+12

a) x+1 b) x-2

Example 9: Given that (x+3) is a factor of x3+3x2-4x-12, find the other linear factors.

October 26, 2017

Example 10: Given that (x-1) is a factor of x3-x2-5x+5, then find the other linear factors.

Example 11: It is know that a polynomial is degree 3 with factors (x-2) and (x+⎷5).

a) What is the 3rd linear factor?

b) What is the polynomial in standard form?

October 26, 2017

Example 12: A polynomial is degree 4 with two zeros at 3⎷2 and -⎷6.

a)What are the other two zeros?

b) Write the polynomial as a product of linear factors.

Square roots factors will always come in pairs as conjugates of each other.

(x-⎷a) and (x+⎷a)

October 26, 2017

Example 13: Given that (x+5) is a factor of x3+5x2+4x+20, then find the other linear factors.

Example 14: A polynomial is degree 4 with two zeros at 4, 2, and 3i.

a)What is the other zero?

b) Write the polynomial as a product of linear factors.

October 26, 2017

c) Write the polynomial in standard form.

Example 15: Write x3-8 as a product of linear factors.

October 26, 2017