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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: 3*x*^{8} + 5*x*^{5} + 2*x*^{3} + *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: 7*x* + 2*x*. 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: **7 x + 2x = 7 + 2 = 9x.**

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

(21*x*^{4} + *x*^{3} + 5*x* - 6) + (-5*x*^{4} + 3*x*^{2} - 5*x* + 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 *x*^{4} so we take 21*x*^{4} from the first string and the -5*x*^{4} from the second string and get:

21*x*^{4} - 5*x*^{4} = 16*x*^{4}

The next highest exponent is *x*^{3} and since there is no “like” monomial, it remains as it is. The same is true for *x*^{2} in 3*x*^{2}. As there is no “like” monomial, it too remains as it is. The 5*x* in the first string and the -5*x* in the second string are like monomials and since they cancel out, i.e., 5*x* - 5*x* = 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:

(21*x*^{4} + *x*^{3} + 5*x* - 6) + (-5*x*^{4} + 3*x*^{2} - 5*x* + 9)

21*x*^{4} - 5*x*^{4} = 16*x*^{4}

*x*^{3}

3*x*^{2}

-6 + 9 = 3

** Solution:** 16

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

(-3*x*^{2} + 8*x* - 2) - (-4*x*^{2} + 6*x* - 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., - (-4

Now the two polynomial strings will read as follows:

(-3*x*^{2} + 8*x* - 2) + (4*x*^{2} - 6*x* + 5)

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

(-3*x*^{2} + 8*x* - 2) - (-4*x*^{2} + 6*x* - 5)

(-3*x*^{2} + 8*x* - 2) + (4*x*^{2} - 6*x* + 5)

-3*x*^{2} + 4*x*^{2} = *x*^{2}

8 - 6 = 2*x*

-2 + 5 = 3

__Solution__:*x*^{2} + 2*x* + 3

1.

(20*x*^{5} + *x* + 19) + (13*x*^{4} + 7*x* - 24)

33*x*^{9} + 8*x* - 5

20*x*^{5} + 13*x*^{4} + 8*x* + 5

33*x*^{9} + 8*x*^{2} - 5

20*x*^{5} + 13*x*^{4} + 8*x* - 5

2.

(7*x*^{6} + 3*x*^{4} - 5*x*^{2} + 23) + (5*x*^{6} + 4*x*^{2} - 14*x* + 33)

12*x*^{6} + 3*x*^{4} + 9*x*^{2} - 14*x* + 56

12*x*^{6} + 3*x*^{4} - *x*^{2} + 56

12*x*^{6} + 3*x*^{4} - *x*^{2} - 14*x* + 56

12*x*^{6} + 3*x*^{4} - *x*^{2} - 14*x* - 10

(7*x*^{6} + 3*x*^{4} - 5*x*^{2} + 23) + (5*x*^{6} + 4*x*^{2} - 14*x* + 33)

7*x*^{6} + 5*x*^{6} = 12*x*^{6}

3*x*^{4}

-5*x*^{2} + 4*x*^{2} = -*x*^{2}

-14*x*

23 + 33 = 56

__Solution__: 12*x*^{6} + 3*x*^{4} - *x*^{2} - 14*x* + 56

Answer (c) is the correct solution

7

3

-5

-14

23 + 33 = 56

Answer (c) is the correct solution

3.

(9*x*^{7} + 2*x*^{4} + 44) + (67*x*^{7} + 51*x*^{4} + 1)

76*x*^{7} + 53*x*^{4} + 43

76*x*^{7} + 49*x*^{4} + 45

76*x*^{14} + 53*x*^{8} + 45

76*x*^{7} + 53*x*^{4} + 45

(9*x*^{7} + 2*x*^{4} + 44) + (67*x*^{7} + 51*x*^{4} + 1)

9*x*^{7} + 67*x*^{7} = 76*x*^{7}

2*x*^{4} + 51*x*^{4} = 53*x*^{4}

44 + 1 = 45

__Solution__: 76*x*^{7} + 53*x*^{4} + 45

Answer (d) is the correct solution

9

2

44 + 1 = 45

Answer (d) is the correct solution

4.

(9*x*^{6} + 5*x*^{4} - 3*x*^{2} + 4) - (-2*x*^{6} + 7*x*^{4} - 12*x* + 99)

11*x*^{6} - 2*x*^{4} - 3*x*^{2} + 12*x* - 95

7*x*^{6} - 12*x*^{4} + 9*x* - 103

7*x*^{6} - 2*x*^{4} + 9*x* - 95

11*x*^{6} + 2*x*^{4} + 12*x* - 95

(9*x*^{6} + 5*x*^{4} - 3*x*^{2} + 4) - (-2*x*^{6} + 7*x*^{4} - 12*x* + 99)

(9*x*^{6} + 5*x*^{4} - 3*x*^{2} + 4) + (2*x*^{6} - 7*x*^{4} + 12*x* - 99)

9*x*^{6} + 2*x*^{6} = 11*x*^{6}

5*x*^{4} - 7*x*^{4} = -2*x*^{4}

-3*x*^{2}

12*x*

4 - 99 = -95

__Solution__: 11*x*^{6} - 2*x*^{4} - 3*x*^{2} + 12*x* - 95

Answer (a) is the correct solution

(9

9

5

-3

12

4 - 99 = -95

Answer (a) is the correct solution

5.

(4*x*^{5} - 6*x*^{2} + 27) - (-13*x*^{5} + 2*x*^{2} + 20)

-9*x*^{5} - 4*x*^{2} + 47

17*x*^{5} - 4*x*^{2} + 47

17*x*^{5} - 8*x*^{2} + 7

9*x*^{5} + 4*x*^{2} + 7

(4*x*^{5} - 6*x*^{2} + 27) - (-13*x*^{5} + 2*x*^{2} + 20)

(4*x*^{5} - 6*x*^{2} + 27) + (13*x*^{5} - 2*x*^{2} - 20)

4*x*^{5} + 13*x*^{5} = 17*x*^{5}

-6*x*^{2} - 2*x*^{2} = -8*x*^{2}

27 - 20 = 7

__Solution__: 17*x*^{5} - 8*x*^{2} + 7

Answer (c) is the correct solution

(4

4

-6

27 - 20 = 7

Answer (c) is the correct solution

6.

(22*x*^{5} + 16*x*^{4} + 11*x* - 31) + (8*x*^{4} + 9*x*^{2} + 35*x* + 42)

30*x*^{5} + 24*x*^{4} + 9*x*^{2} + 46*x* + 11

22*x*^{5} + 24*x*^{4} + 9*x*^{2} + 46*x* + 11

30*x*^{9} + 24*x*^{8} + 9*x*^{2} + 46*x* + 11

22*x*^{5} + 8*x*^{4} + 9*x*^{2} + 46*x* + 11

(22*x*^{5} + 16*x*^{4} + 11*x* - 31) + (8*x*^{4} + 9*x*^{2} + 35*x* + 42)

22*x*^{5}

16*x*^{4} + 8*x*^{4} = 24*x*^{4}

9*x*^{2}

11*x* + 35*x* = 46*x*

-31 + 42 = 11

__Solution__: 22*x*^{5} + 24*x*^{4} + 9*x*^{2} + 46*x* + 11

Answer (b) is the correct solution

22

16

9

11

-31 + 42 = 11

Answer (b) is the correct solution

7.

(4*x*^{3} + 2*x*^{2} - 6*x* + 18) - (-2*x*^{3} + 3*x*^{2} - 39*x* - 16)

6*x*^{3} - *x*^{2} + 33*x* + 34

2*x*^{3} + 5*x*^{2} - 33*x* + 2

6*x*^{3} - 5*x*^{2} + 33*x* + 34

2*x*^{3} - *x*^{2} - 33*x* + 34

(4*x*^{3} + 2*x*^{2} - 6*x* + 18) - (-2*x*^{3} + 3*x*^{2} - 39*x* - 16)

(4*x*^{3} + 2*x*^{2} - 6x + 18) + (2*x*^{3} - 3*x*^{2} + 39*x* + 16)

4*x*^{3} + 2*x*^{3} = 6*x*^{3}

2*x*^{2} - 3*x*^{2} = -*x*^{2}

-6*x* + 39*x* = 33*x*

18 + 16 = 34

__Solution__: 6*x*^{3} - *x*^{2} + 33*x* + 34

Answer (a) is the correct solution

(4

4

2

-6

18 + 16 = 34

Answer (a) is the correct solution

8.

(12*x*^{4} + 9*x* - 15) - (6*x*^{4} + 8*x* - 10)

18*x*^{4} + 17*x* - 25

6*x*^{4} + *x* - 5

6*x*^{4} + 17*x* - 25

18*x*^{4} + *x* + 5

(12*x*^{4} + 9*x* - 15) - (6*x*^{4} + 8*x* - 10)

(12*x*^{4} + 9*x* - 15) + (-6*x*^{4} - 8*x* + 10)

12*x*^{4} - 6*x*^{4} = 6*x*^{4}

9*x* - 8*x* = *x*

-15 + 10 = -5

__Solution__: 6*x*^{4} + *x* - 5

Answer (b) is the correct solution

(12

12

9

-15 + 10 = -5

Answer (b) is the correct solution

9.

(-6*x*^{2} + 7*x* - 9) + (6*x*^{2} - 5*x* + 13)

2*x* + 4

(-6*x*^{2} + 7*x* - 9) + (6*x*^{2} - 5*x* + 13)

7*x* - 5*x* = 2*x*

-9 + 13 = 4

__Solution__: 2*x* + 4

Answer (c) is the correct solution. [__Note__: As (-6*x* and 6*x* cancel each other out, they do not need to appear in the solution.]

7

-9 + 13 = 4

Answer (c) is the correct solution. [

10.

(48*x*^{6} + 32*x*^{3} + 22) - (9*x*^{6} + 16*x*^{3} - 4)

39*x*^{6} + 16*x*^{3} + 18

39*x*^{6} + 16*x*^{3} + 26

57*x*^{6} + 48*x*^{3} + 18

57*x*^{6} + 16*x*^{3} + 26

(48*x*^{6} + 32*x*^{3} + 22) - (9*x*^{6} + 16*x*^{3} - 4)

(48*x*^{6} + 32*x*^{3} + 22) + (-9*x*^{6} - 16*x*^{3} + 4)

48*x*^{6} - 9*x*^{6} = 39*x*^{6}

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

22 + 4 = 26

__Solution__: 39*x*^{6} + 16*x*^{3} + 26

Answer (b) is the correct solution

(48

48

32

22 + 4 = 26

Answer (b) is the correct solution

x^{5}+x+ 19) + (13x^{4}+ 7x- 24)20

x^{5}13

x^{4}x+ 7x= 8x19 - 24 = -5

Solution: 20x^{5}+ 13x^{4}+ 8x- 5Answer (d) is the correct solution