Adding Polynomials

Background

Polynomials are expressions of finite length constructed from variables.  Note that the only operations used are addition, subtraction, and non-negative whole number exponents. For example, we know that 2x³ − 7x + 10 is a polynomial but x² − 3/x + 7x^7/2     is not because the second term involves division by the variable x and because the third term contains a non-integer exponent.

Application

Recall in your elementary years you added numbers horizontally like this:

8 + 3= 11.

We may add polynomials in the same way by combining like terms. Let us take a look at an example:

Simplify (3x + 4y) + (5x – 2y)

First, we may omit the parentheses and combine like terms.

(3x + 4y) + (5x – 2y)  
  =  3x + 5y + 5x – 2y  
  =  3x + 5x + 4y – 2y  
  =  8x + 2y

An alternative to adding horizontally is to add vertically. Looking back at the previous example, here

        Simplify (3x + 4y) + (5x – 2y)

Each variable shall be placed in its own column. The first column will be the x-column, and the second column will be the y-column:

                          3x + 4y

                          5x – 2y

                          8x + 2y

Note that we get the same solution vertically as we got horizontally.

Practice Problem

Now, let us do a practice problem

Simplify (4x³ + 3x² – 2x + 7) + (x³ – 2x² + x – 4)

Solution

Horizontal Method:

(4x³ + 3x² – 2x + 7) + (x³ – 2x² + x – 4) 

=  4x³ + 3x² – 2x + 7 + x³ – 2x² + x – 4  
  =  4x³ + x³ + 3x² – 2x² – 2x + x + 7 – 4  
  =  5x³ + 1x² – 1x + 3

Vertical Method

4x³ + 3x² – 2x + 7

x³ – 2x² + x – 4

5x³ + 1x² – 1x + 3