 # Math: Middle School: Grades 6, 7 and 8 Quiz - Pre-Algebra - Polynomials: Add and Subtract (Questions)

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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). For example: 3x8 + 5x5 + 2x3 + x. This example has four monomials that are linked by addition signs (+) making it a polynomial.

When writing out a polynomial string as the example shown above, the general rule is to list each monomial in a descending order based upon the exponents. In the above example, the exponents are 8, 5, 3 and 1. An “x” that does not show an exponent is considered to have the exponent of “1”.

When adding or subtracting two or more polynomial strings the first thing to do is to look for the “like” terms (or monomial) in each string. (Note: When you have more than one string, each string is surrounded by parentheses ().)

What are “like” terms? Like terms are when you have the same variable (x, y) and/or the same exponents. For example: 7x + 2x. Here the variable “x” is the same so then you can easily work the coefficients, i.e., “7” and “2” or 7 + 2 = 9. They each have the same variable “x” so the problem is worked: 7x + 2x = 7 + 2 = 9x.

Now let’s look at adding two polynomial strings and see what we need to do to work them out.

(21x4 + x3 + 5x - 6) + (-5x4 + 3x2 - 5x + 9)

Remember to work in descending order so locate the “like” monomials with the highest exponent first. In this case the like monomial with the highest exponent is x4 so we take 21x4 from the first string and the -5x4 from the second string and get:

21x4 - 5x4 = 16x4

The next highest exponent is x3 and since there is no “like” monomial, it remains as it is. The same is true for x2 in 3x2. As there is no “like” monomial, it too remains as it is. The 5x in the first string and the -5x in the second string are like monomials and since they cancel out, i.e., 5x - 5x = 0, they are no longer needed in the solution. That then leaves us with two constant numbers, i.e., -6 and 9. -6 + 9 = 3.

The full way to write out the solution for our polynomial strings is:

(21x4 + x3 + 5x - 6) + (-5x4 + 3x2 - 5x + 9)
21x4 - 5x4 = 16x4
x3
3x2
-6 + 9 = 3
Solution: 16x4 + x3 + 3x2 + 3

Now let’s look at subtracting two polynomial strings and see what we need to do to work them out.

(-3x2 + 8x - 2) - (-4x2 + 6x - 5)

[Note: When you have to subtract a negative number you have to add the opposite of that number. So let’s look at our second string above, i.e., - (-4x2 + 6x - 5). As you have to add the opposite number, this string will change to look like: + (4x2 - 6x + 5). Notice that each mathematical sign changed to the opposite.]

Now the two polynomial strings will read as follows:

(-3x2 + 8x - 2) + (4x2 - 6x + 5)

From here the solution is worked out the same as it was for addition so the solution is worked as follows:

(-3x2 + 8x - 2) - (-4x2 + 6x - 5)
(-3x2 + 8x - 2) + (4x2 - 6x + 5)
-3x2 + 4x2 = x2
8 - 6 = 2x
-2 + 5 = 3
Solution: x2 + 2x + 3

1. (-6x2 + 7x - 9) + (6x2 - 5x + 13)
 [ ] x2 + 2x + 4 [ ] x2 + 2x + 22 [ ] 2x + 4 [ ] x2 + 12x + 22
2. (48x6 + 32x3 + 22) - (9x6 + 16x3 - 4)
 [ ] 39x6 + 16x3 + 18 [ ] 39x6 + 16x3 + 26 [ ] 57x6 + 48x3 + 18 [ ] 57x6 + 16x3 + 26
3. (9x6 + 5x4 - 3x2 + 4) - (-2x6 + 7x4 - 12x + 99)
 [ ] 11x6 - 2x4 - 3x2 + 12x - 95 [ ] 7x6 - 12x4 + 9x - 103 [ ] 7x6 - 2x4 + 9x - 95 [ ] 11x6 + 2x4 + 12x - 95
4. (9x7 + 2x4 + 44) + (67x7 + 51x4 + 1)
 [ ] 76x7 + 53x4 + 43 [ ] 76x7 + 49x4 + 45 [ ] 76x14 + 53x8 + 45 [ ] 76x7 + 53x4 + 45
5. (22x5 + 16x4 + 11x - 31) + (8x4 + 9x2 + 35x + 42)
 [ ] 30x5 + 24x4 + 9x2 + 46x + 11 [ ] 22x5 + 24x4 + 9x2 + 46x + 11 [ ] 30x9 + 24x8 + 9x2 + 46x + 11 [ ] 22x5 + 8x4 + 9x2 + 46x + 11
6. (4x5 - 6x2 + 27) - (-13x5 + 2x2 + 20)
 [ ] -9x5 - 4x2 + 47 [ ] 17x5 - 4x2 + 47 [ ] 17x5 - 8x2 + 7 [ ] 9x5 + 4x2 + 7
7. (20x5 + x + 19) + (13x4 + 7x - 24)
 [ ] 33x9 + 8x - 5 [ ] 20x5 + 13x4 + 8x + 5 [ ] 33x9 + 8x2 - 5 [ ] 20x5 + 13x4 + 8x - 5
8. (4x3 + 2x2 - 6x + 18) - (-2x3 + 3x2 - 39x - 16)
 [ ] 6x3 - x2 + 33x + 34 [ ] 2x3 + 5x2 - 33x + 2 [ ] 6x3 - 5x2 + 33x + 34 [ ] 2x3 - x2 - 33x + 34
9. (7x6 + 3x4 - 5x2 + 23) + (5x6 + 4x2 - 14x + 33)
 [ ] 12x6 + 3x4 + 9x2 - 14x + 56 [ ] 12x6 + 3x4 - x2 + 56 [ ] 12x6 + 3x4 - x2 - 14x + 56 [ ] 12x6 + 3x4 - x2 - 14x - 10
10. (12x4 + 9x - 15) - (6x4 + 8x - 10)
 [ ] 18x4 + 17x - 25 [ ] 6x4 + x - 5 [ ] 6x4 + 17x - 25 [ ] 18x4 + x + 5 Math: Middle School: Grades 6, 7 and 8 Quiz - Pre-Algebra - Polynomials: Add and Subtract (Answers)
1. (-6x2 + 7x - 9) + (6x2 - 5x + 13)
 [ ] x2 + 2x + 4 [ ] x2 + 2x + 22 [x] 2x + 4 [ ] x2 + 12x + 22
(-6x2 + 7x - 9) + (6x2 - 5x + 13)
7x - 5x = 2x
-9 + 13 = 4
Solution: 2x + 4
Answer (c) is the correct solution. [Note: As (-6x and 6x cancel each other out, they do not need to appear in the solution.]
2. (48x6 + 32x3 + 22) - (9x6 + 16x3 - 4)
 [ ] 39x6 + 16x3 + 18 [x] 39x6 + 16x3 + 26 [ ] 57x6 + 48x3 + 18 [ ] 57x6 + 16x3 + 26
(48x6 + 32x3 + 22) - (9x6 + 16x3 - 4)
(48x6 + 32x3 + 22) + (-9x6 - 16x3 + 4)
48x6 - 9x6 = 39x6
32x3 - 16x3 = 16x3
22 + 4 = 26
Solution: 39x6 + 16x3 + 26
Answer (b) is the correct solution
3. (9x6 + 5x4 - 3x2 + 4) - (-2x6 + 7x4 - 12x + 99)
 [x] 11x6 - 2x4 - 3x2 + 12x - 95 [ ] 7x6 - 12x4 + 9x - 103 [ ] 7x6 - 2x4 + 9x - 95 [ ] 11x6 + 2x4 + 12x - 95
(9x6 + 5x4 - 3x2 + 4) - (-2x6 + 7x4 - 12x + 99)
(9x6 + 5x4 - 3x2 + 4) + (2x6 - 7x4 + 12x - 99)
9x6 + 2x6 = 11x6
5x4 - 7x4 = -2x4
-3x2
12x
4 - 99 = -95
Solution: 11x6 - 2x4 - 3x2 + 12x - 95
Answer (a) is the correct solution
4. (9x7 + 2x4 + 44) + (67x7 + 51x4 + 1)
 [ ] 76x7 + 53x4 + 43 [ ] 76x7 + 49x4 + 45 [ ] 76x14 + 53x8 + 45 [x] 76x7 + 53x4 + 45
(9x7 + 2x4 + 44) + (67x7 + 51x4 + 1)
9x7 + 67x7 = 76x7
2x4 + 51x4 = 53x4
44 + 1 = 45
Solution: 76x7 + 53x4 + 45
Answer (d) is the correct solution
5. (22x5 + 16x4 + 11x - 31) + (8x4 + 9x2 + 35x + 42)
 [ ] 30x5 + 24x4 + 9x2 + 46x + 11 [x] 22x5 + 24x4 + 9x2 + 46x + 11 [ ] 30x9 + 24x8 + 9x2 + 46x + 11 [ ] 22x5 + 8x4 + 9x2 + 46x + 11
(22x5 + 16x4 + 11x - 31) + (8x4 + 9x2 + 35x + 42)
22x5
16x4 + 8x4 = 24x4
9x2
11x + 35x = 46x
-31 + 42 = 11
Solution: 22x5 + 24x4 + 9x2 + 46x + 11
Answer (b) is the correct solution
6. (4x5 - 6x2 + 27) - (-13x5 + 2x2 + 20)
 [ ] -9x5 - 4x2 + 47 [ ] 17x5 - 4x2 + 47 [x] 17x5 - 8x2 + 7 [ ] 9x5 + 4x2 + 7
(4x5 - 6x2 + 27) - (-13x5 + 2x2 + 20)
(4x5 - 6x2 + 27) + (13x5 - 2x2 - 20)
4x5 + 13x5 = 17x5
-6x2 - 2x2 = -8x2
27 - 20 = 7
Solution: 17x5 - 8x2 + 7
Answer (c) is the correct solution
7. (20x5 + x + 19) + (13x4 + 7x - 24)
 [ ] 33x9 + 8x - 5 [ ] 20x5 + 13x4 + 8x + 5 [ ] 33x9 + 8x2 - 5 [x] 20x5 + 13x4 + 8x - 5
(20x5 + x + 19) + (13x4 + 7x - 24)
20x5
13x4
x + 7x = 8x
19 - 24 = -5
Solution: 20x5 + 13x4 + 8x - 5
Answer (d) is the correct solution
8. (4x3 + 2x2 - 6x + 18) - (-2x3 + 3x2 - 39x - 16)
 [x] 6x3 - x2 + 33x + 34 [ ] 2x3 + 5x2 - 33x + 2 [ ] 6x3 - 5x2 + 33x + 34 [ ] 2x3 - x2 - 33x + 34
(4x3 + 2x2 - 6x + 18) - (-2x3 + 3x2 - 39x - 16)
(4x3 + 2x2 - 6x + 18) + (2x3 - 3x2 + 39x + 16)
4x3 + 2x3 = 6x3
2x2 - 3x2 = -x2
-6x + 39x = 33x
18 + 16 = 34
Solution: 6x3 - x2 + 33x + 34
Answer (a) is the correct solution
9. (7x6 + 3x4 - 5x2 + 23) + (5x6 + 4x2 - 14x + 33)
 [ ] 12x6 + 3x4 + 9x2 - 14x + 56 [ ] 12x6 + 3x4 - x2 + 56 [x] 12x6 + 3x4 - x2 - 14x + 56 [ ] 12x6 + 3x4 - x2 - 14x - 10
(7x6 + 3x4 - 5x2 + 23) + (5x6 + 4x2 - 14x + 33)
7x6 + 5x6 = 12x6
3x4
-5x2 + 4x2 = -x2
-14x
23 + 33 = 56
Solution: 12x6 + 3x4 - x2 - 14x + 56
Answer (c) is the correct solution
10. (12x4 + 9x - 15) - (6x4 + 8x - 10)
 [ ] 18x4 + 17x - 25 [x] 6x4 + x - 5 [ ] 6x4 + 17x - 25 [ ] 18x4 + x + 5
(12x4 + 9x - 15) - (6x4 + 8x - 10)
(12x4 + 9x - 15) + (-6x4 - 8x + 10)
12x4 - 6x4 = 6x4
9x - 8x = x
-15 + 10 = -5
Solution: 6x4 + x - 5
Answer (b) is the correct solution