Mathematics » Polynomials II » Add and Subtract Polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

## Example

Find the sum: $$\left(5{y}^{2}-3y+15\right)+\left(3{y}^{2}-4y-11\right).$$

### Solution

 Identify like terms.  Rearrange to get the like terms together.  Combine like terms.  ## Example

Find the difference: $$\left(9{w}^{2}-7w+5\right)-\left(2{w}^{2}-4\right).$$

### Solution  Distribute and identify like terms.  Rearrange the terms.  Combine like terms.  ## Example

Subtract: $$\left({c}^{2}-4c+7\right)$$ from $$\left(7{c}^{2}-5c+3\right)$$.

### Solution    Distribute and identify like terms.  Rearrange the terms.  Combine like terms.  ## Example

Find the sum: $$\left({u}^{2}-6uv+5{v}^{2}\right)+\left(3{u}^{2}+2uv\right)$$.

### Solution

$$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\left({u}^{2}-6uv+5{v}^{2}\right)+\left(3{u}^{2}+2uv\right)\hfill \\ \text{Distribute.}\hfill & & & \phantom{\rule{4em}{0ex}}{u}^{2}-6uv+5{v}^{2}+3{u}^{2}+2uv\hfill \\ \text{Rearrange the terms, to put like terms together.}\hfill & & & \phantom{\rule{4em}{0ex}}{u}^{2}+3{u}^{2}-6uv+2uv+5{v}^{2}\hfill \\ \text{Combine like terms.}\hfill & & & \phantom{\rule{4em}{0ex}}4{u}^{2}-4uv+5{v}^{2}\hfill \end{array}$$

## Example

Find the difference: $$\left({p}^{2}+{q}^{2}\right)-\left({p}^{2}+10pq-2{q}^{2}\right)$$.

### Solution

$$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\left({p}^{2}+{q}^{2}\right)-\left({p}^{2}+10pq-2{q}^{2}\right)\hfill \\ \text{Distribute.}\hfill & & & \phantom{\rule{4em}{0ex}}{p}^{2}+{q}^{2}-{p}^{2}-10pq+2{q}^{2}\hfill \\ \text{Rearrange the terms, to put like terms together.}\hfill & & & \phantom{\rule{4em}{0ex}}{p}^{2}-{p}^{2}-10pq+{q}^{2}+2{q}^{2}\hfill \\ \text{Combine like terms.}\hfill & & & \phantom{\rule{4em}{0ex}}-10p{q}^{2}+3{q}^{2}\hfill \end{array}$$

## Example

Simplify: $$\left({a}^{3}-{a}^{2}b\right)-\left(a{b}^{2}+{b}^{3}\right)+\left({a}^{2}b+a{b}^{2}\right)$$.

### Solution

$$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\left({a}^{3}-{a}^{2}b\right)-\left(a{b}^{2}+{b}^{3}\right)+\left({a}^{2}b+a{b}^{2}\right)\hfill \\ \text{Distribute.}\hfill & & & \phantom{\rule{4em}{0ex}}{a}^{3}-{a}^{2}b-a{b}^{2}-{b}^{3}+{a}^{2}b+a{b}^{2}\hfill \\ \text{Rearrange the terms, to put like terms together.}\hfill & & & \phantom{\rule{4em}{0ex}}{a}^{3}-{a}^{2}b+{a}^{2}b-a{b}^{2}+a{b}^{2}-{b}^{3}\hfill \\ \text{Combine like terms.}\hfill & & & \phantom{\rule{4em}{0ex}}{a}^{3}-{b}^{3}\hfill \end{array}$$