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Divide Polynomials
Objectives:
1. To divide polynomials using long division and synthetic division
Warm-Up
Use long division to divide 5 into 3462.
5 34626
30-46
9
45-12
2
10-2
Warm-Up
Use long division to divide 5 into 3462.
5 34626
30-46
9
45-12
2
10-2
Divisor Dividend
Quotient
Remainder
Warm-Up
Use long division to divide 5 into 3462.
3462 2692
5 5
Dividend
Divisor
Quotient
Remainder
Divisor
Remainders
If you are lucky enough to get a remainder of zero when dividing, then the divisor divides evenly into the dividend
This means that the divisor is a factor of 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:
If your polynomial is missing a term, add it in with a coefficient of 0 as a place holder
2 𝑥3+𝑥+32 𝑥3+0 𝑥2+𝑥+3
Exercise 1
Divide x + 1 into x2 + 3x + 5
Line up the first term of the quotient with the term of the dividend with the same degree.
21 3 5x x x
How many times does x go into x2?
x
Multiply x by x + 12 x x-
2x-
5
2
Multiply 2 by x + 12 2x - -3
Exercise 1
Divide x + 1 into x2 + 3x + 5
21 3 5x x x x
2 x x-2x-
5
2
2 2x - -3
Divisor
Dividend
Quotient
Remainder
Exercise 1
Divide x + 1 into x2 + 3x + 5
2 3 5 32
1 1
x xx
x x
Divisor
Dividend
Quotient
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. 2.
Exercise 5
Use long division to divide x4 – 10x2 + 2x + 3 by x – 3
Objective 1bYou 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.
k a b c d
a
ka
= Add terms
= Multiply by k
Coefficients of Quotient (in decreasing order)
Remainder
Synthetic Division
Synthetic Division (of a Cubic Polynomial)
To divide ax3 + bx2 + cx + d by x – k, use the following pattern.
k a b c d
a
ka
= Add terms
= Multiply by k
You are always adding columns using synthetic division, whereas you subtracted columns in long division.
Synthetic Division (of a Cubic Polynomial)
To divide ax3 + bx2 + cx + d by x – k, use the following pattern.
Synthetic Division
You are always adding columns using synthetic division, whereas you subtracted columns in long division.
k can be positive or negative. If you divide by x + 2, then k = -2 because
x + 2 = x – (-2).
Add a coefficient of zero for any missing terms!
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. 2.
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
The volume of the solid is 3x3 + 8x2 – 45x – 50. Find an expression for the missing dimension.
x +
5
? x + 1
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.
Divide Polynomials
Objectives:
1. To divide polynomials using long division and synthetic division
