Pre-Algebra - Polynomials: Division (Part 1)

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Pre-Algebra - Polynomials: Division (Part 1)

This Math quiz is called 'Pre-Algebra - Polynomials: Division (Part 1)' and it has been written by teachers to help you if you are studying the subject at middle school. Playing educational quizzes is a fabulous way to learn if you are in the 6th, 7th or 8th grade - aged 11 to 14.

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A polynomial is two or more monomial numbers that are linked together in an equation by an addition sign (+), subtraction sign (-) or a multiplication sign (x). In this quiz, we will be dealing with dividing a polynomial by a monomial - a number that stands on its own.

Let’s look at the following example of a polynomial divided by a monomial.

15x2 + 10x2 + 5x ÷ 5x [Note: This may also be written as (15x2 + 10x2 + 5x) ÷ (5x).] The numbers in the first set of parentheses is the polynomial and the number in the second set of parentheses is the monomial.

Now let’s work the problem.

(15x2 + 10x2 + 5x) ÷ (5x)
15x2 ÷ 5x = 3x
10x2 ÷ 5x = 2x
5x ÷ 5x = 1
Solution: 3x + 2x + 1

[Note: When you divide each of the numbers by the monomial, the exponent numbers are subtracted from each other so in the example problem above, looking at 15x2 ÷ 5x, the variables x2 and x are subtracted so that you have x2 - x (which is understood to be x1) = x which is why the sum of 15x2 ÷ 5x is 3x. And as 5x ÷ 5x would be x - x = 0, the 5x ÷ 5x = 1 because the variables equaled to “0” so the variable is cancelled out.] So if we were to write out each step in the above problem, it would be worked as follows:

(15x2 + 10x2 + 5x) ÷ (5x)
15x2 ÷ 5x = (15 ÷ 5 = 3) and (x2 - x = x)
15x2 ÷ 5x = 3x
10x2 ÷ 5x = (10 ÷ 5 = 2) and (x2 - x = x)
10x2 ÷ 5x = 2x
5x ÷ 5x = (5 ÷ 5 = 1) and (x - x = 0)
5x ÷ 5x = 1
Solution: 3x + 2x + 1

As long as you remember that the exponents are subtracted, there is no need to write out each of the steps listed directly above but you can write them out as listed in the first example. However, this longer version should help you to understand the process.

1.
(96x2 - 36x2) ÷ (12x)
8x3 - 3x3 = 5x3
8x + 3x = 11x
8x - 3x = 5x
8x3 + 3x3 = 11x3
(96x2 - 36x2) ÷ (12x)
96x2 ÷ 12x = 8x
-36x2 ÷ 12x = -3x
Solution: 8x - 3x = 5x
Answer (c) is the correct solution. (Did you remember to subtract your exponents?) In addition, this problem has a final answer of 5x as 8x and 3x are “like” numbers
2.
(50x6 + 30x5 - 20x) ÷ (10x)
5x5 + 3x4 - 2x2
5x5 + 3x4 - 2x
5x5 + 3x4 + 2
5x5 + 3x4 - 2
(50x6 + 30x5 - 20x) ÷ (10x)
50x6 ÷ 10x = 5x5
30x5 ÷ 10x = 3x4
-20x ÷ 10x = -2
Solution: 5x5 + 3x4 - 2
Answer (d) is the correct solution. (Did you remember to subtract your exponents?)
3.
(99x9 + 72x4 + 36x2) ÷ (9x)
11x10 + 8x5 + 4x3
11x8 + 8x3 + 4x
11x8 + 8x3 + 4x3
11x9 + 8x4 + 4x2
(99x9 + 72x4 + 36x2) ÷ (9x)
99x9 ÷ 9x = 11x8
72x4 ÷ 9x = 8x3
36x2 ÷ 9x = 4x
Solution: 11x8 + 8x3 + 4x
Answer (b) is the correct solution. (Did you remember to subtract your exponents?)
4.
(57x6 + 111x5 - 15x4) ÷ (3x4)
19x2 + 37x + 5
19x2 + 37x - 5x
19x2 + 37x - 5
19x2 + 37x + 5x
(57x6 + 111x5 - 15x4) ÷ (3x4)
57x6 ÷ 3x4 = 19x2
111x5 ÷ 3x4 = 37x
-15x4 ÷ 3x4 = -5
Solution: 19x2 + 37x - 5
Answer (c) is the correct solution. (Did you remember to subtract your exponents?)
5.
(45x5 + 35x3) ÷ (-5x)
(-9x6) + (-7x4)
9x6 + 7x4
9x4 + 7x2
(-9x4) + (-7x2)
(45x5 + 35x3) ÷ (-5x)
45x5 ÷ (-5x) = -9x4
35x3 ÷ (-5x) = -7x2
Solution: (-9x4) + (-7x2)
(In order to not confuse anyone between the divide symbol and the negative value of 5, parentheses are placed around the negative value monomial.)
Answer (d) is the correct solution. (Did you remember to subtract your exponents?)
6.
x3 + 18x2 + 12x) ÷ (6x)
4x2 + 3x2 + 2
4x2 + 3x2 + 2x
4x2 + 3x + 2
4x + 3x + 2x
(24x3 + 18x2 + 12x) ÷ (6x)
24x3 ÷ 6x = 4x2
18x2 ÷ 6x = 3x
12x ÷ 6x = 2
Solution: 4x2 + 3x + 2
Answer (c) is the correct solution. (Did you remember to subtract your exponents?)
7.
(28x6 + 21x3 + 14x) ÷ (7x)
4x6 + 3x3 + 2x
4x5 + 3x2 + 2
4x5 + 3x + 2
4x + 3x2 + 2x
(28x6 + 21x3 + 14x) ÷ (7x)
28x6 ÷ 7x = 4x5
21x3 ÷ 7x = 3x2
14x ÷ 7x = 2
Solution: 4x5 + 3x2 + 2
Answer (b) is the correct solution. (Did you remember to subtract your exponents?)
8.
(64x8 + 40x6) ÷ (8x3)
8x5 + 5x3
8x5 + 5x3 + 8x3
8x11 + 5x9
8x24 + 5x18
(64x8 + 40x6) ÷ (8x3)
64x8 ÷ 8x3 = 8x5
40x6 ÷ 8x3 = 5x3
Solution: 8x5 + 5x3
Answer (a) is the correct solution. (Did you remember to subtract your exponents?)
9.
(62x8 + 28x5) ÷ (-2x2)
(-31x6) + (-14x3)
31x6 + 14x3
(31x6) - (14x3)
(-31x10) + (-14x7)
(62x8 + 28x5) ÷ (-2x2)
62x8 ÷ (-2x2) = -31x6
28x5 ÷ (-2x2) = -14x3
Solution: (-31x6) + (-14x3)
(In order to not confuse anyone between the divide symbol and the negative value of 2, parentheses are placed around the negative value monomial.)
Answer (a) is the correct solution. (Did you remember to subtract your exponents?)
10.
(78x2 + 52x2) ÷ (13x2)
6x2 + 4x2
6 + 4 = 10
6x2 + 4x2 = 10x4
6x4 + 4x4 = 10x8
(78x2 + 52x2) ÷ (13x2)
78x2 ÷ 13x2 = 6
52x2 ÷ 13x2 = 4
Solution: 6 + 4 = 10
Answer (b) is the correct solution. (Did you remember to subtract your exponents?) In addition, this problem has a final answer of 10 as 6 and 4 are “like” numbers
Author:  Christine G. Broome

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