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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. 24*x*^{2} is also a monomial. (__ Note:__ the “24” is a

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: 2*x*^{2} + 6*x*^{2}.

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., 4*x*^{7} + 5*x*^{6} - 9*x*^{5} 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 3*x*, the *x* is understood to have an exponent of “^{1}”. So if we were to **add** the 3*x* to our polynomial it would be added in descending order to read: 4*x*^{7} + 5*x*^{6} - 9*x*^{5} + 3*x*.

What are “like” terms? Like terms are when you have the same variable and/or the same exponents. For example: 4*x* - 6*x*. 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: **4 x - 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: 5*x*^{3} - 2*x*^{3} + 10. This would be worked out as: 5*x*^{3} - 2*x*^{3} + 10 = 5 - 2 = 3. As the same variable is *x*^{3} then the problem would proceed to be written as: 5*x*^{3} - 2*x*^{3} + 10 = 5 - 2 = 3*x*^{3} + 10. Since 3*x*^{3} and 10 are __NOT__ like numbers, you are done working the problem so the sum or answer of this polynomial is **3 x^{3} + 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.

43*x* + *x* =

43*x* + *x* = 43 + 1 = 44*x*^{2}

43*x* + *x* = 43 + 1 = 44

43*x* + *x* = 43 + 1 = 44*x*

43*x* + *x* = 43 + 1 = 44 - 2*x*

2.

2*x*^{2} + 8*x*^{2} =

2*x*^{2} + 8*x*^{2} = 2 + 8 = 10*x*^{4}

2*x*^{2} + 8*x*^{2} = 2 + 8 = 10*x*^{2}

2*x*^{2} + 8*x*^{2} = 16*x*^{2}

2*x*^{2} + 8*x*^{2} = 2 - 8 = 10*x*^{2}

2*x*^{2} + 8*x*^{2} = 2 + 8 = 10

as the variables are like variables, i.e.,*x*^{2} the problem is worked as

2*x*^{2} + 8*x*^{2} = 2 + 8 = 10*x*^{2}

Answer (b) shows the correct result of working the problem

as the variables are like variables, i.e.,

2

Answer (b) shows the correct result of working the problem

3.

12*x*^{4} - 10*x*^{4} + 82 =

2*x*^{8} + 82

2*x*^{16} + 82

2*x* + 82

2*x*^{4} + 82

12*x*^{4} - 10*x*^{4} + 82 = 12 - 10 = 2

As the variables are like variables, i.e.,*x*^{4} the problem is worked as

12*x*^{4} - 10*x*^{4} + 82 = 12 - 10 = 2*x*^{4} + 82

The answer to this polynomial is 2*x*^{4} + 82. Answer (d) is correct

As the variables are like variables, i.e.,

12

The answer to this polynomial is 2

4.

Whenever you simply have an “

As the variables are like variables, i.e.,

Answer (c) shows the correct result of working the problem

5.

13*x*^{6} + 27*x*^{6} =

13*x*^{6} + 27*x*^{6} = 13 + 27 = 40*x*^{12}

13*x*^{6} + 27*x*^{6} = 13 + 27 = 40*x*^{36}

13*x*^{6} + 27*x*^{6} = 13 + 27 = 40*x*^{6}

13*x*^{6} + 27*x*^{6} = 13 + 27 = 40

13*x*^{6} + 27*x*^{6} = 13 + 27 = 40

as the variables are like variables, i.e.,*x*^{6} the problem is worked as

13*x*^{6} + 27*x*^{6} = 13 + 27 = 40*x*^{6}

as the variables are like variables, i.e.,

13

6.

36*x*^{3} - 16*x*^{3} =

36*x*^{3} - 16*x*^{3} = 36 - 16 = 20*x*^{3}

36*x*^{3} - 16*x*^{3} = 36 - 16 = 20*x*^{6}

36*x*^{3} - 16*x*^{3} = 36 - 16 = 20*x*^{9}

36*x*^{3} - 16*x*^{3} = 36 - 16 = 20*x*

36*x*^{3} - 16*x*^{3} = 36 - 16 = 20

as the variables are like variables, i.e.,*x*^{3} the problem is worked as

36*x*^{3} - 16*x*^{3} = 36 - 16 = 20*x*^{3}

Answer (a) shows the correct result of working the problem

as the variables are like variables, i.e.,

36

Answer (a) shows the correct result of working the problem

7.

4*x*^{6} + 7*x*^{6} + 33 =

44*x*^{6}

11*x*^{6} + 33

44*x*^{36}

11*x*^{12} + 33

4*x*^{6} + 7*x*^{6} + 33 = 4 + 7 = 11

As the variables are like variables, i.e.,*x*^{6} the problem is worked as

4*x*^{6} + 7*x*^{6} + 33 = 4 + 7 = 11*x*^{6} + 33.

The answer to this polynomial is 11*x*^{6} + 33. Answer (b) is correct

As the variables are like variables, i.e.,

4

The answer to this polynomial is 11

8.

7 + 5*x*^{4} - 8 + 4*x*^{3} + 3*x*^{4} + 15 =

12*x*^{11} + 14

8*x*^{4} + 4*x*^{3} + 14

8*x*^{4} + 4*x*^{3} + 30

32*x*^{7} + 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 *x*^{4}. Work the like terms out as 5*x*^{4} + 3*x*^{4} = 5 + 3 = 8*x*^{4}. The next exponent in descending order is 4*x*^{3}. As there is no other like number with *x*^{3}, the problem should now read: 8*x*^{4} + 4*x*^{3}. The remaining numbers, i.e., 7, -8 and 15 are then worked. 7 - 8 = -1 + 15 = 14. The final worked answer should now read: 8*x*^{4} + 4*x*^{3} + 14. Answer (b) is correct

9.

-1 + 6*x*^{2} - 4 - 7*x*^{2} + *x*^{6} + 9 =

5*x*^{2} - 11*x*^{2} + *x*^{6} + 9

-6*x*^{2} + *x*^{6} + 9

4 + 13*x*^{2} + *x*^{6}

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, *x*^{6} (understood to mean 1*x*^{6}.) The next like term in descending order are the two *x*^{2}. They are worked out as: 6*x*^{2} - 7*x*^{2} = 6 - 7 = -*x*^{2} (understood to be -1*x*^{2}). At this point the worked problem should look like: *x*^{6} - *x*^{2}. 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: *x*^{6} - *x*^{2} + 4. Answer (a) is correct

10.

9*x*^{5} - 3*x*^{5} + 14 =

6*x*^{10} + 14

6*x*^{25} + 14

20*x*^{5}

6*x*^{5} + 14

9*x*^{5} - 3*x*^{5} + 14 = 9 - 3 = 6

As the variables are like variables, i.e.,*x*^{5} the problem is worked as

9*x*^{5} - 3*x*^{5} + 14 = 9 - 3 = 6*x*^{5} + 14

The answer to this polynomial is 6*x*^{5} + 14. Answer (d) is correct

As the variables are like variables, i.e.,

9

The answer to this polynomial is 6

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x+x= 43 + 1 = 44Whenever 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.,xthe problem is worked as43

x+x= 43 + 1 = 44xAnswer (c) shows the correct result of working the problem