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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.5 y squared minus 3 y plus 15, plus 3 y squared minus 4 y minus 11.
Rearrange to get the like terms together.5y squared plus 3y squared, identified as like terms, minus 3y minus 4y, identified as like terms, plus 15 minus 11, identified as like terms.
Combine like terms.8 y squared minus 7y plus 4.

Example

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

Solution

 9 w squared minus 7 w plus 5, minus 2 w squared minus 4.
Distribute and identify like terms.9 w squared and 2 w squared are like terms. 5 and 4 are also like terms.
Rearrange the terms.9 w squared minus 2 w squared minus 7 w plus 5 plus 4.
Combine like terms.7 w squared minus 7 w plus 9.

Example

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

Solution

 .
 7 c squared minus 5 c plus 3, minus c squared minus 4c plus 7.
Distribute and identify like terms.7 c squared and c squared are like terms. Minus 5c and 4c are like terms. 3 and minus 7 are like terms.
Rearrange the terms.7 c squared minus c squared minus 5 c plus 4 c plus 3 minus 7.
Combine like terms.6 c squared minus c minus 4.

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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