Addition and Subtraction of Polynomials

To find the sum of two or more polynomials, we use the associative and commutative properties of addition to group like terms together, and the we combine the like terms by using the distributive property.

Example:

Find the following sum: (2{x^2} + 7x - 5) + (3{x^2} - 11x + 8)

Solution:
(2{x^2} + 7x - 5) + (3{x^2} - 11x + 8)
 = (2{x^2} + 3{x^2}) + (7x - 11x) + ( - 5 + 8)
 = 5{x^2} - 4x + 3

Example:

Find the following difference: (3{x^3} - 5{x^2} + 8x - 3) - (5{x^3} - 7x + 11)

Solution:
(3{x^3} - 5{x^2} + 8x - 3) - (5{x^3} - 7x + 11)
 = (3{x^3} - 5{x^2} + 8x - 3) + ( - 5{x^3} + 7x - 11)
 = (3{x^3} - 5{x^3}) - 5{x^2} + (8x + 7x) + ( - 3 - 11)
 = - 2{x^3} - 5{x^2} + 15x - 14

In adding or subtracting polynomials, you may prefer to use a vertical arrangement with like terms in the same columns.

Example:

Perform the indicated operations:

(4{x^3} + 7{x^2}y + 2x{y^2} - 2{y^3}) + (2{x^3} + x{y^2} + 4{y^3}) + (4{x^2}y - 8x{y^2} - 9{y^3}) - ({y^3} - 7{x^2}y)

Solution:

First we use a vertical arrangement to perform the addition.

\begin{array}{*{20}{c}} {}&{}&{4{x^3}}& + &{7{x^2}y}& + &{2x{y^2}}& - &{2{y^3}} \\ {( + )}&{}&{2{x^3}}&{}&{}& + &{x{y^2}}& + &{4{y^3}} \\ {( + )}&{}&{}&{}&{4{x^2}y}& - &{8x{y^2}}& - &{9{y^3}} \end{array}


\overline {\begin{array}{*{20}{c}} {}&{}&{6{x^3}}& + &{11{x^2}y}& - &{5x{y^2}}& - &{7{y^3}} \end{array}}

Then we perform the subtraction vertically.

\frac{{\begin{array}{*{20}{c}} {}&{}&{6{x^3}}& + &{11{x^2}y}& - &{5x{y^2}}& - &{7{y^3}} \\ {( - )}&{}&{}& - &{7{x^2}y}&{}&{}& + &{{y^3}} \end{array}}}{{\begin{array}{*{20}{c}} {}&{}&{6{x^3}}& + &{18{x^2}y}& - &{5x{y^2}}& - &{8{y^3}} \end{array}}}