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# Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, $5{x}^{2}$ and $-2{x}^{2}$ are like terms, and can be added to get $3{x}^{2}$, but $3x$ and $3{x}^{2}$ are not like terms, and therefore cannot be added.

### How To: Given multiple polynomials, add or subtract them to simplify the expressions.

1. Combine like terms.
2. Simplify and write in standard form.

### Example 2: Adding Polynomials

Find the sum.

$\left(12{x}^{2}+9x - 21\right)+\left(4{x}^{3}+8{x}^{2}-5x+20\right)$

### Solution

$\begin{array}{cc}4{x}^{3}+\left(12{x}^{2}+8{x}^{2}\right)+\left(9x - 5x\right)+\left(-21+20\right) \hfill & \text{Combine like terms}.\hfill \\ 4{x}^{3}+20{x}^{2}+4x - 1\hfill & \text{Simplify}.\hfill \end{array}$

### Analysis of the Solution

We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.

### Try It 2

Find the sum. [latex-display]\left(2{x}^{3}+5{x}^{2}-x+1\right)+\left(2{x}^{2}-3x - 4\right)[/latex-display] Solution

### Example 3: Subtracting Polynomials

Find the difference.

$\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)$

### Solution

[latex-display]\begin{array}{cc}7{x}^{4}-5{x}^{3}+\left(-{x}^{2}+2{x}^{2}\right)+\left(6x - 3x\right)+\left(1 - 2\right)\text{ }\hfill & \text{Combine like terms}.\hfill \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\hfill & \text{Simplify}.\hfill \end{array}[/latex-display]

### Analysis of the Solution

Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.

### Try It 3

Find the difference. [latex-display]\left(-7{x}^{3}-7{x}^{2}+6x - 2\right)-\left(4{x}^{3}-6{x}^{2}-x+7\right)[/latex-display] Solution