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# Pre-Algebra - Polynomials

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In algebra there are several terms to understand in order to work a problem. One term is the monomial. “Mono” refers to “one” so a monomial is a mathematical number that stands alone. For example: 24 is a monomial. 24x2 is also a monomial. (Note: the “24” is a coefficient and the “x” is a variable [unknown number] and the 2 is an exponent. Exponents are also referred to the “degree” of a number or term. So the degree of 24x2 is 2.)

For each polynomial problem given below, work the polynomial problem to show the sum or answer of the polynomial (watching for like numbers).

1.
2x2 + 8x2 =
2x2 + 8x2 = 2 + 8 = 10x4
2x2 + 8x2 = 2 + 8 = 10x2
2x2 + 8x2 = 16x2
2x2 + 8x2 = 2 - 8 = 10x2
2x2 + 8x2 = 2 + 8 = 10
as the variables are like variables, i.e., x2 the problem is worked as
2x2 + 8x2 = 2 + 8 = 10x2
Answer (b) shows the correct result of working the problem
2.
7 + 5x4 - 8 + 4x3 + 3x4 + 15 =
12x11 + 14
8x4 + 4x3 + 14
8x4 + 4x3 + 30
32x7 + 14
To work this problem, you must first look for the “like” numbers or terms. Remember, when writing the sum or answer, the numbers or terms are placed in descending order so you need to find the highest exponent to work first. In this case the like term is x4. Work the like terms out as 5x4 + 3x4 = 5 + 3 = 8x4. The next exponent in descending order is 4x3. As there is no other like number with x3, the problem should now read: 8x4 + 4x3. The remaining numbers, i.e., 7, -8 and 15 are then worked. 7 - 8 = -1 + 15 = 14. The final worked answer should now read: 8x4 + 4x3 + 14. Answer (b) is correct
3.
36x3 - 16x3 =
36x3 - 16x3 = 36 - 16 = 20x3
36x3 - 16x3 = 36 - 16 = 20x6
36x3 - 16x3 = 36 - 16 = 20x9
36x3 - 16x3 = 36 - 16 = 20x
36x3 - 16x3 = 36 - 16 = 20
as the variables are like variables, i.e., x3 the problem is worked as
36x3 - 16x3 = 36 - 16 = 20x3
Answer (a) shows the correct result of working the problem
4.
12x4 - 10x4 + 82 =
2x8 + 82
2x16 + 82
2x + 82
2x4 + 82
12x4 - 10x4 + 82 = 12 - 10 = 2
As the variables are like variables, i.e., x4 the problem is worked as
12x4 - 10x4 + 82 = 12 - 10 = 2x4 + 82
The answer to this polynomial is 2x4 + 82. Answer (d) is correct
5.
9x5 - 3x5 + 14 =
6x10 + 14
6x25 + 14
20x5
6x5 + 14
9x5 - 3x5 + 14 = 9 - 3 = 6
As the variables are like variables, i.e., x5 the problem is worked as
9x5 - 3x5 + 14 = 9 - 3 = 6x5 + 14
The answer to this polynomial is 6x5 + 14. Answer (d) is correct
6.
4x6 + 7x6 + 33 =
44x6
11x6 + 33
44x36
11x12 + 33
4x6 + 7x6 + 33 = 4 + 7 = 11
As the variables are like variables, i.e., x6 the problem is worked as
4x6 + 7x6 + 33 = 4 + 7 = 11x6 + 33.
The answer to this polynomial is 11x6 + 33. Answer (b) is correct
7.
43x + x =
43x + x = 43 + 1 = 44x2
43x + x = 43 + 1 = 44
43x + x = 43 + 1 = 44x
43x + x = 43 + 1 = 44 - 2x
43x + x = 43 + 1 = 44
Whenever you simply have an “x” that is standing alone, it is understood to be the same as 1x. As the variables are like variables, i.e., x the problem is worked as
43x + x = 43 + 1 = 44x
Answer (c) shows the correct result of working the problem
8.
-1 + 6x2 - 4 - 7x2 + x6 + 9 =
x6 - x2 + 4
5x2 - 11x2 + x6 + 9
-6x2 + x6 + 9
4 + 13x2 + x6
To work this problem, you must first look for the “like” numbers or terms. Remember, when writing the sum or answer, the numbers or terms are placed in descending order so you need to find the highest exponent to work first. In this case, x6 (understood to mean 1x6.) The next like term in descending order are the two x2. They are worked out as: 6x2 - 7x2 = 6 - 7 = -x2 (understood to be -1x2). At this point the worked problem should look like: x6 - x2. The remaining numbers, i.e., -1, -4 and 9 are then worked. -1 and -4 equal -5 + 9 = 4. The final worked answer should now read: x6 - x2 + 4. Answer (a) is correct
9.
13x6 + 27x6 =
13x6 + 27x6 = 13 + 27 = 40x12
13x6 + 27x6 = 13 + 27 = 40x36
13x6 + 27x6 = 13 + 27 = 40x6
13x6 + 27x6 = 13 + 27 = 40
13x6 + 27x6 = 13 + 27 = 40
as the variables are like variables, i.e., x6 the problem is worked as
13x6 + 27x6 = 13 + 27 = 40x6
10.
x - 7x =
x - 7x = 1 - 7 = 6x
x - 7x = 0 - 7 = -7x
x - 7x = 1 - 7 = -6x
x - 7x = 0 + 7 = 7x
x - 7x = 1 - 7 = -6
Whenever you simply have an “x” that is standing alone, it is understood to be the same as 1x. However, 1x is rarely ever written out as such so just a simple x is used.
As the variables are like variables, i.e., x the problem is worked as
x - 7x = 1 - 7 = -6x
Answer (c) shows the correct result of working the problem
Author:  Christine G. Broome