When adding and subtracting polynomials , you can use the distributive property to add or subtract the coefficients of like terms.

Example 1:

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

Use the commutative property to group like terms.

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

Use the distributive property.

$\begin{array}{l}=\left(2+3\right){x}^{2}+\left(5-2\right)x+\left(7+5\right)\\ =5{x}^{2}+3x+12\end{array}$

(Recall that "like terms" are monomials with the same variables, such as $3{x}^{2}y$ and $82{x}^{2}y$ .)

It may be helpful to arrange the like terms in columns, as with multi-digit addition.

Example 2:

Subtract.

$\left({x}^{2}+3xy-9\right)-\left(-2{y}^{2}+5xy+6\right)$

Arrange vertically.

$\begin{array}{l}\begin{array}{cccccccc}& {x}^{2}& & & +& 3xy& -& 9\\ -& & -& 2{y}^{2}& +& 5xy& +& 6\end{array}\hfill \\ \stackrel{¯}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}\hfill \end{array}$

Subtract term by term.

$\begin{array}{l}\begin{array}{cccccccc}& {x}^{2}& & & +& 3xy& -& 9\\ -& & -& 2{y}^{2}& +& 5xy& +& 6\end{array}\hfill \\ \stackrel{¯}{\begin{array}{llllllll}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\hfill & {x}^{2}\hfill & +\hfill & 2{y}^{2}\hfill & -\hfill & 2xy\hfill & -\hfill & 15\hfill \end{array}}\hfill \end{array}$ .

The solution is ${x}^{2}+2{y}^{2}-2xy-15$ .