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 6.7

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

Solution
Identify like terms. \(\left(\underline{\underline{5 y^{2}}}-\underline{3 y}+\underline{15}\right)+\left(\underline{\underline{3 y^{2}}}-\underline{4 y}-\underline{11}\right)\)
Rearrange to get the like terms together. \(\underline{5 y^{2}+3 y^{2}}-\underline{3 y-4 y}+\underline{15}-\underline{11}\)
Combine like terms. \(8 y^{2}-7 y+4\)
Try It 6.13

Find the sum: \(\left(7 x^{2}-4 x+5\right)+\left(x^{2}-7 x+3\right)\).

Try It 6.14

Find the sum: \(\left(14 y^{2}+6 y-4\right)+\left(3 y^{2}+8 y+5\right)\).

Example 6.8

Find the difference: \((9w^2−7w+5)−(2w^2−4)\).

Solution
  \(\left(9 w^{2}-7 w+5\right)-\left(2 w^{2}-4\right)\)
Distribute and identify like terms. \(\underline{\underline{9 w^{2}}}-\underline{7 w}+\underset{\sim}{5}-\underline{\underline{2 w^{2}}}+\underset{\sim}{4}\)
Rearrange the terms. \(\underline{\underline{9 w^{2}-2 w^{2}}}-\underline{7 w} + \underset{\sim}{5}+\underset{\sim}{4}\)
Combine like terms. \(7 w^{2}-7 w+9\)
Try It 6.15

Find the difference: \(\left(8 x^{2}+3 x-19\right)-\left(7 x^{2}-14\right)\).

Try It 6.16

Find the difference: \(\left(9 b^{2}-5 b-4\right)-\left(3 b^{2}-5 b-7\right)\).

Example 6.9

Subtract: \((c^2−4c+7) from (7c^2−5c+3)\).

Solution
  Subtract \(\left(c^{2}-4 c+7\right)\) from \(\left(7 c^{2}-5 c+3\right)\).
  \(\left(7 c^{2}-5 c+3\right)-\left(c^{2}-4 c+7\right)\)
Distribute and identify like terms. \(\underline{\underline{7 c^{2}}}-\underline{5 c}+\underset{\sim}{3}-\underline{\underline{c^{2}}}+\underline{4 c}-\underset{\sim}{7}\)
Rearrange the terms. \(\underline{\underline{7 c^{2}-c^{2}}}-\underline{5 c+4 c}+\underset{\sim}{3}-\underset{\sim}{7}\)
Combine like terms. \(6 c^{2}-c-4\)
Try It 6.17

Subtract: \(\left(5 z^{2}-6 z-2\right)\) from \(\left(7 z^{2}+6 z-4\right)\).

Try It 6.18

Subtract: \(\left(x^{2}-5 x-8\right)\) from \(\left(6 x^{2}+9 x-1\right)\).

Example 6.10

Find the sum: \((u^2−6uv+5v^2)+(3u^2+2uv)\).

Solution
  \(\left(u^{2}-6 u v+5 v^{2}\right)+\left(3 u^{2}+2 u v\right)\)
Distribute. \(u^{2}-6 u v+5 v^{2}+3 u^{2}+2 u v\)
Rearrange the terms, to put like terms together. \(u^{2}+3 u^{2}-6 u v+2 u v+5 v^{2}\)
Combine like terms. \(4 u^{2}-4 u v+5 v^{2}\)
Try It 6.19

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

Try It 6.20

Find the sum: \(\left(2 x^{2}-3 x y-2 y^{2}\right)+\left(5 x^{2}-3 x y\right)\).

Example 6.11

Find the difference: \((p^2+q^2)−(p^2+10pq−2q^2)\).

Solution
  \(\left(p^{2}+q^{2}\right)-\left(p^{2}+10 p q-2 q^{2}\right)\)
Distribute. \(p^{2}+q^{2}-p^{2}-10 p q+2 q^{2}\)
Rearrange the terms, to put like terms together. \(p^{2}-p^{2}-10 p q+q^{2}+2 q^{2}\)
Combine like terms. \(-10 p q^{2}+3 q^{2}\)
Try It 6.21

Find the difference: \(\left(a^{2}+b^{2}\right)-\left(a^{2}+5 a b-6 b^{2}\right)\).

Try It 6.22

Find the difference: \(\left(m^{2}+n^{2}\right)-\left(m^{2}-7 m n-3 n^{2}\right)\).

Example 6.12

Simplify: \((a^3−a^2b)−(ab^2+b^3)+(a^2b+ab^2)\).

Solution
  \(\left(a^{3}-a^{2} b\right)-\left(a b^{2}+b^{3}\right)+\left(a^{2} b+a b^{2}\right)\)
Distribute. \(a^{3}-a^{2} b-a b^{2}-b^{3}+a^{2} b+a b^{2}\)
Rearrange the terms, to put like terms together. \(a^{3}-a^{2} b+a^{2} b-a b^{2}+a b^{2}-b^{3}\)
Combine like terms. \(a^{3}-b^{3}\)
Try It 6.23

Simplify: \(\left(x^{3}-x^{2} y\right)-\left(x y^{2}+y^{3}\right)+\left(x^{2} y+x y^{2}\right)\).

Try It 6.24

Simplify: \(\left(p^{3}-p^{2} q\right)+\left(p q^{2}+q^{3}\right)-\left(p^{2} q+p q^{2}\right)\).

Back to '8.2: Adding and Subtracting Polynomials'