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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

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

1.

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.

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

3.

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

4.

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

5.

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

6.

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

7.

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*

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

Whenever you simply have an “*x*” that is standing alone, it is understood to be the same as 1*x*. As the variables are like variables, i.e., *x* the problem is worked as

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

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

Whenever you simply have an “

43

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

8.

-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

9.

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

10.

Whenever you simply have an “

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

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

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x^{2}+ 8x^{2}= 2 + 8 = 10as the variables are like variables, i.e.,

x^{2}the problem is worked as2

x^{2}+ 8x^{2}= 2 + 8 = 10x^{2}Answer (b) shows the correct result of working the problem