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Adding and Subtracting Polynomials

Adding and Subtracting 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

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

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