Pre-Algebra - Polynomials: Add and Subtract

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.

(21x⁴ + x³ + 5x - 6) + (-5x⁴ + 3x² - 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 x⁴ so we take 21x⁴ from the first string and the -5x⁴ from the second string and get:

21x⁴ - 5x⁴ = 16x⁴

The next highest exponent is x³ and since there is no “like” monomial, it remains as it is. The same is true for x² in 3x². 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:

(21x⁴ + x³ + 5x - 6) + (-5x⁴ + 3x² - 5x + 9)
21x⁴ - 5x⁴ = 16x⁴
x³
3x²
-6 + 9 = 3
Solution: 16x⁴ + x³ + 3x² + 3

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

(-3x² + 8x - 2) - (-4x² + 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., - (-4x² + 6x - 5). As you have to add the opposite number, this string will change to look like: + (4x² - 6x + 5). Notice that each mathematical sign changed to the opposite.]

Now the two polynomial strings will read as follows:

(-3x² + 8x - 2) + (4x² - 6x + 5)

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

(-3x² + 8x - 2) - (-4x² + 6x - 5)
(-3x² + 8x - 2) + (4x² - 6x + 5)
-3x² + 4x² = x²
8 - 6 = 2x
-2 + 5 = 3
Solution: x² + 2x + 3