5.5ish: Divide Polynomials

5.5ish: Divide Polynomials
1.
Objectives:
To divide
polynomials using
long division and
synthetic division
Assignment:
• P. 366-368: 2-20, 36,
37, 47, 48
• Challenge Problems
Warm-Up
Use long division to divide 5 into 3462.
6 92
5 3462
- 30
46
- 45
12
- 10
2
Warm-Up
Use long division to divide 5 into 3462.
Divisor
6 92
5 3462
- 30
46
- 45
12
- 10
2
Quotient
Dividend
Remainder
Warm-Up
Use long division to divide 5 into 3462.
Dividend
Divisor
3462
2
 692 
5
5
Quotient
Remainder
Divisor
Remainders
This means that the divisor is a
factor of the dividend
If you are lucky enough to get a
remainder of zero when dividing, then the
divisor divides evenly into the dividend
For example, when dividing 3 into 192, the
remainder is 0. Therefore, 3 is a factor of 192.
Vocabulary
Quotient
Remainder
Dividend
Divisor
Divides Evenly
Factor
Objective 1a
You will be able to
divide polynomials
using long division
Dividing Polynomials
Dividing polynomials works just like long
division. In fact, it is called long
division!
Before you start dividing:
Make sure the
divisor and dividend
are in standard form
If your polynomial is
missing a term, add it in
with a coefficient of 0 as
a place holder
Dividing Polynomials
Dividing polynomials works just like long
division. In fact, it is called long
division!
Before you start dividing:
2𝑥 3 + 𝑥 + 3
2𝑥 3 + 0𝑥 2 + 𝑥 + 3
If your polynomial is
missing a term, add it in
with a coefficient of 0 as
a place holder
Exercise 1
Divide x + 1 into x2 + 3x + 5
x 2
x  1 x 2  3x  5
- x2  - x
2x  5
- 2 x  -2
3
How many times
does x go into x2?
Multiply x by x + 1
Multiply 2 by x + 1
Line up the first term of the quotient with the
term of the dividend with the same degree.
Exercise 1
Divide x + 1 into x2 + 3x + 5
x 2
x  1 x 2  3x  5
- x2  - x
2x  5
- 2 x  -2
Divisor
3
Quotient
Dividend
Remainder
Exercise 1
Divide x + 1 into x2 + 3x + 5
Dividend
x 2  3x  5
3
 x2
x 1
x 1
Quotient
Divisor
Remainder
Divisor
Exercise 2
Divide 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5
Exercise 3
In a polynomial division problem, if the
degree of the dividend is m and the degree
of the divisor is n, what is the degree of the
quotient?
Exercise 4
Divide using long division.
1.
𝑥 3 −𝑥 2 +4𝑥−10
𝑥+2
2.
2𝑥 4 +𝑥 3 +𝑥−1
𝑥 2 +2𝑥−1
Exercise 5
Use long division to divide x4 – 10x2 + 2x + 3
by x – 3
Objective 1b
You will be able to
divide polynomials
using synthetic
division
Synthetic Division
When you divisor is of the form x  k,
where k is a constant, then you can
perform the division quicker and easier
using just the coefficients of the dividend.
This is called fake
division. I mean,
synthetic division.
Synthetic Division
Synthetic Division (of a Cubic Polynomial)
To divide ax3 + bx2 + cx + d by x – k, use the
following pattern.
= Add terms
k a b
c
d
ka
a
= Multiply by k
Remainder
Coefficients of Quotient (in decreasing order)
Synthetic Division
Synthetic Division (of a Cubic Polynomial)
To divide ax3 + bx2 + cx + d by x – k, use the
following pattern.
= Add terms
k a b
c
d
ka
= Multiply by k
a
You are always adding columns using synthetic division,
whereas you subtracted columns in long division.
Synthetic Division
Synthetic Division (of a Cubic Polynomial)
To divide ax3 + bx2 + cx + d by x – k, use the
following pattern.
k can be positive or
Add a coefficient
of zero for any
missing terms!
negative. If you divide by
x + 2, then k = -2 because
x + 2 = x – (-2).
You are always adding columns using synthetic division,
whereas you subtracted columns in long division.
Exercise 6
Use synthetic division to divide
x4 – 10x2 + 2x + 3 by x – 3
Exercise 7
Divide 2x3 + 9x2 + 4x + 5 by x + 3 using
synthetic division
Exercise 8
Divide using long division.
1.
𝑥 3 +4𝑥 2 −𝑥−1
𝑥+3
2.
4𝑥 3 +𝑥 2 −3𝑥+7
𝑥−1
Exercise 9
Given that x – 4 is a factor of
x3 – 6x2 + 5x + 12, rewrite
x3 – 6x2 + 5x + 12 as a product of two
polynomials.
Exercise 10
x+5
The volume of the solid
is 3x3 + 8x2 – 45x – 50.
Find an expression for
the missing
dimension.
?
Exercise 11
Use long division to divide
6x4 – 11x3 + 14x2 – 3x – 1 by 2x – 1. Then
figure out a way to perform the division
synthetically.
5.5ish: Divide Polynomials
1.
Objectives:
To divide
polynomials using
long division and
synthetic division
Assignment
• P. 366-368: 2-20,
36, 37, 47, 48
• Challenge
Problems
“Does this have something to do with the Factor and Remainder Theorems?”