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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.)

A polynomial refers to more than one monomial or, in other words, it refers to two or more monomial numbers that are linked together through an equation by an addition sign (+), subtraction sign (-) or a multiplication sign (x). “Poly” refers to “many.” For example: 2x2 + 6x2.

This example has two monomials that are linked by an addition sign (+) making it a polynomial.

Although this example is a polynomial, generally when you have only two monomials linked, it is called a binomial.

When you have three monomials linked, i.e., 4x7 + 5x6 - 9x5 then you have what is referred to as a trinomial. Anything beyond three linked monomials is referred to as a polynomial even though both the binomial and the trinomial can also be referred to as polynomials. (Note: When you have a polynomial, the degree of the polynomial is the highest exponent number so in this polynomial the highest exponent is “7” making the degree of the entire polynomial “7”.)

When writing out polynomials, they are generally written in the descending order of exponents. For example, let’s relook at the above polynomial. The exponents are 7, 6 and 5 and are written in a descending order. Whenever you have a number such as 3x, the x is understood to have an exponent of “1”. So if we were to add the 3x to our polynomial it would be added in descending order to read: 4x7 + 5x6 - 9x5 + 3x.

What are “like” terms? Like terms are when you have the same variable and/or the same exponents. For example: 4x - 6x. Here the variable “x” is the same so then you can easily work the coefficients, i.e., “4” and “6” or 4 - 6 = -2. They each have the same variable “x” so the problem is worked: 4x - 6x = 4 - 6 = -2x.

When working a problem with like terms, once you have collected your “like” terms, you are done with the problem. For example, take the polynomial: 5x3 - 2x3 + 10. This would be worked out as: 5x3 - 2x3 + 10 = 5 - 2 = 3. As the same variable is x3 then the problem would proceed to be written as: 5x3 - 2x3 + 10 = 5 - 2 = 3x3 + 10. Since 3x3 and 10 are NOT like numbers, you are done working the problem so the sum or answer of this polynomial is 3x3 + 10.

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

1.
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
2.
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
3.
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
4.
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
5.
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
6.
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
7.
-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
8.
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
9.
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
10.
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
Author:  Christine G. Broome

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