dividing-polynomials-worksheet

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I. Model Problems.
II. Practice

III. Challenge Problems
VI. Answer Key

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I. Model Problems

A monomial is an expression that is a number, variable or product of a
number and variables.
Examples of monomials: –3, 4x, 5xy, y2

A polynomial is a monomial or the sum or difference of monomials.
Examples of polynomials: 2x + 4, –x4 + 4x3 – 5x2, 400

To divide a polynomial by a monomial, multiply each term in the
polynomial by the reciprocal of the monomial. Add the results.

Example 1 Divide 6x3 + 4x2 – 10x + 14 by 2x2

= (6x3 + 4x2 – 10x + 14)• 1 Multiply by the reciprocal of the
2x2 divisor.
Distributive property.
= 6x3  4x2  10x  14
2x2 2x2 2x2 2x2 Simplify.

= 3x + 2 – 5 + 7
x x2

The answer is 3x + 2 – 5+ 7
.
x x2

To divide a polynomial by another polynomial, use polynomial long

division. This process is similar to long division using numbers.

Recall that to divide numbers you can use the following long division

algorithm:

41 First, divide 14 into 57; it goes in 4 times. Multiply 4 by

14 575 14. Write the product (56) under 57. Subtract 56 from
56 57 to get 1. Drop down the 5. Divide 14 into 15; it goes
15 in 1 time. Multiply 14 by 1. Write the product (14)
under 15. Subtract 14 from 15. 1 (the difference) is the

14 remainder.

1

Polynomial long division is shown in the example below:

Example 2 Divide (6x2 – 7x – 5) by (2x + 1).

3x  5 First divide 2x into 6x2; the quotient is 3x because
2x 1 6x2  7x  5 2x(3x) = 6x2. Write 3x on top. Multiply 3x by 2x + 1
and write the product 6x2 + 3x beneath the dividend.
6x2  3x Subtract from 6x2 – 7x to obtain –10x. Drop down
 10x  5
the -5. Divide 2x into -10x; the quotient is -5. Write -
  10x  5 5 on top. Multiply 5 by 2x + 1 to obtain 10x – 5.
Subtract from -10x – 5. The remainder is 0.
0

The answer is (3x – 5).

The polynomial in the dividend must contain a term for each degree. If
there is no term for a degree, use a placeholder. This is shown in the
example below.

Example 3 Divide (5x3 – 7x – 1) by (x – 1).
5x2  5x  2

x  1 5x3  0x2  7x  1
5x3  5x2
5x2  7x
 5x2  5x
 2x 1
  2x  2
3

The answer is 5x2 + 5x – 2, remainder -3.
This can also be written as 5x2 + 5x – 2  3 .

x1

II. Practice

Divide.

1. (8x2 + 4x)  2x 2. (16x3 + 4x + 20)  4x

3. (30x3 + 50x)  5x 4. (12x3 + 16x2 – 8x)  2x

5. (27x3 + 12x)  3x2 6. (36x4 – 48x)  –6x

7. (x2 + 7x + 10)  (x + 2) 8. (x2 – 3x – 54)  (x + 9)

9. (x2 + 6x + 9)  (x + 3) 10. (x2 – 25)  (x + 5)

11. (3x3 + 11x2 + 9x – 5)  (x + 2) 12. (x2 + 5x + 8)  (x + 2)

13. (x2 – 10,000)  (x – 100) 14. (x3 + 9x2 + x + 9)  (x + 9)

15. (2x2 + 3x – 65)  (2x + 13) 16. (12m2 + 57m + 66) 
(4m + 11)

17. (x4 – 5x3 – 7x2 + 36x – 5)  18. (–x2 – 94x + 600)  (x + 100)
(x – 5) 20. (4x3 – 8x + 3)  (x + 2)

19. (6x2 + 5x – 50)  (2x – 5)

21. (x3 + 7x2 + 16x + 12)  22. (y3 – 3y2 – 17y + 3)  (y + 3)

(x + 2) 24. (y3 – 2y2 – 45y – 50)  (y + 5)
23. (x3 + 10x2 + 32x + 33) 
26. (12y3 + 27y2 + 75y + 65) 
(x + 3) (3y + 3)
25. (2x3 – 11x2 – 2x + 2) 

(2x + 1)

27. (15x3 + 37x2 + 53x + 55)  28. (16y3 + 44y2 – 40y + 7) 
(3x + 5) (2y + 7)

III. Challenge Problems

29. The area of a rectangle is given by the expression
(x3 + 7x2 + 11x + 2) in2. The length of the rectangle is given by the
expression (x + 2) in. What is an expression for the width of the
rectangle?
_________________________________________________________

_________________________________________________________

30. Explain how polynomial long division is similar to and different
from arithmetic long division.

_________________________________________________________

_________________________________________________________

31. Correct the Error
There is an error in the student work shown below:
Question: Divide (x3 – 8)  (x – 2).
Solution:

x2
x 2 x3 8

x 3  2x 2
6  2x 2
The answer is x2 remainder -6 + 2x2.

What is the error? Explain how to solve the problem.
_________________________________________________________

_________________________________________________________

IV. Answer Key

1. 4x + 2
2. 4x2 + 1 + 5/x
3. 6x2 + 10
4. 6x2 + 8x – 4

5. 9x + 4/x
6. –6x3 + 8

7. x + 5
8. x – 6

9. x + 3
10. x – 5
11. 3x2 + 5x – 1 remainder -3

12. x + 3 remainder 2

13. (x + 100)
14. x2 + 1
15. (x – 5)

16. (3m + 6)
17. (x3 – 7x + 1)
18. (–x + 6)

19. (3x + 10)
20. (4x2 – 8x + 8) remainder -13
21. (x2 + 5x + 6)
22. (y2 – 6y + 1)
23. (x2 + 7x + 11)
24. (y2 – 7y – 10)
25. (x2 – 6x + 2)
26. (4y2 + 5y + 20) remainder 5
27. (5x2 + 4x + 11)
28. (8y2 – 6y + 1)
29. (x2 + 5x + 1) in.

30. Answers will vary; both processes use the same algorithm that

involves dividing, multiplying and subtracting.
31. The student did not include placeholders between x3 and -8. The
dividend should be x3 + 0x2 + 0x – 8.