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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: 3x^{8} + 5x^{5} + 2x^{3} + x. This example has four monomials that are linked by addition signs (+) making it a polynomial.
[readmore]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: 7x + 2x. 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: 7x + 2x = 7 + 2 = 9x.
Now let’s look at adding two polynomial strings and see what we need to do to work them out.
(21x^{4} + x^{3} + 5x  6) + (5x^{4} + 3x^{2}  5x + 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 21x^{4} from the first string and the 5x^{4} from the second string and get:
21x^{4}  5x^{4} = 16x^{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 3x^{2}. As there is no “like” monomial, it too remains as it is. The 5x in the first string and the 5x in the second string are like monomials and since they cancel out, i.e., 5x  5x = 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:
(21x^{4} + x^{3} + 5x  6) + (5x^{4} + 3x^{2}  5x + 9)
21x^{4}  5x^{4} = 16x^{4}
x^{3}
3x^{2}
6 + 9 = 3
Solution: 16x^{4} + x^{3} + 3x^{2} + 3
Now let’s look at subtracting two polynomial strings and see what we need to do to work them out.
(3x^{2} + 8x  2)  (4x^{2} + 6x  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.,  (4x^{2} + 6x  5). As you have to add the opposite number, this string will change to look like: + (4x^{2}  6x + 5). Notice that each mathematical sign changed to the opposite.]
Now the two polynomial strings will read as follows:
(3x^{2} + 8x  2) + (4x^{2}  6x + 5)
From here the solution is worked out the same as it was for addition so the solution is worked as follows:
(3x^{2} + 8x  2)  (4x^{2} + 6x  5)
(3x^{2} + 8x  2) + (4x^{2}  6x + 5)
3x^{2} + 4x^{2} = x^{2}
8  6 = 2x
2 + 5 = 3
Solution: x^{2} + 2x + 3















(48x^{6} + 32x^{3} + 22)  (9x^{6} + 16x^{3}  4)
(48x^{6} + 32x^{3} + 22) + (9x^{6}  16x^{3} + 4) 48x^{6}  9x^{6} = 39x^{6} 32x^{3}  16x^{3} = 16x^{3} 22 + 4 = 26 Solution: 39x^{6} + 16x^{3} + 26 Answer (b) is the correct solution 

(9x^{6} + 5x^{4}  3x^{2} + 4)  (2x^{6} + 7x^{4}  12x + 99)
(9x^{6} + 5x^{4}  3x^{2} + 4) + (2x^{6}  7x^{4} + 12x  99) 9x^{6} + 2x^{6} = 11x^{6} 5x^{4}  7x^{4} = 2x^{4} 3x^{2} 12x 4  99 = 95 Solution: 11x^{6}  2x^{4}  3x^{2} + 12x  95 Answer (a) is the correct solution 
(9x^{7} + 2x^{4} + 44) + (67x^{7} + 51x^{4} + 1)
9x^{7} + 67x^{7} = 76x^{7} 2x^{4} + 51x^{4} = 53x^{4} 44 + 1 = 45 Solution: 76x^{7} + 53x^{4} + 45 Answer (d) is the correct solution 

(22x^{5} + 16x^{4} + 11x  31) + (8x^{4} + 9x^{2} + 35x + 42)
22x^{5} 16x^{4} + 8x^{4} = 24x^{4} 9x^{2} 11x + 35x = 46x 31 + 42 = 11 Solution: 22x^{5} + 24x^{4} + 9x^{2} + 46x + 11 Answer (b) is the correct solution 
(4x^{5}  6x^{2} + 27)  (13x^{5} + 2x^{2} + 20)
(4x^{5}  6x^{2} + 27) + (13x^{5}  2x^{2}  20) 4x^{5} + 13x^{5} = 17x^{5} 6x^{2}  2x^{2} = 8x^{2} 27  20 = 7 Solution: 17x^{5}  8x^{2} + 7 Answer (c) is the correct solution 

(20x^{5} + x + 19) + (13x^{4} + 7x  24)
20x^{5} 13x^{4} x + 7x = 8x 19  24 = 5 Solution: 20x^{5} + 13x^{4} + 8x  5 Answer (d) is the correct solution 
(4x^{3} + 2x^{2}  6x + 18)  (2x^{3} + 3x^{2}  39x  16)
(4x^{3} + 2x^{2}  6x + 18) + (2x^{3}  3x^{2} + 39x + 16) 4x^{3} + 2x^{3} = 6x^{3} 2x^{2}  3x^{2} = x^{2} 6x + 39x = 33x 18 + 16 = 34 Solution: 6x^{3}  x^{2} + 33x + 34 Answer (a) is the correct solution 

(7x^{6} + 3x^{4}  5x^{2} + 23) + (5x^{6} + 4x^{2}  14x + 33)
7x^{6} + 5x^{6} = 12x^{6} 3x^{4} 5x^{2} + 4x^{2} = x^{2} 14x 23 + 33 = 56 Solution: 12x^{6} + 3x^{4}  x^{2}  14x + 56 Answer (c) is the correct solution 
(12x^{4} + 9x  15)  (6x^{4} + 8x  10)
(12x^{4} + 9x  15) + (6x^{4}  8x + 10) 12x^{4}  6x^{4} = 6x^{4} 9x  8x = x 15 + 10 = 5 Solution: 6x^{4} + x  5 Answer (b) is the correct solution 
7x  5x = 2x
9 + 13 = 4
Solution: 2x + 4
Answer (c) is the correct solution. [Note: As (6x and 6x cancel each other out, they do not need to appear in the solution.]