1.10: Adding and Subtracting Polynomial Expressions

Polynomials can be added, subtracted, multiplied and divided. When adding or subtracting, we can only combine terms that are like terms.

What Are Like Terms?

Consider the expression

\[5 x^{4} y^{2}+6 x^{3} y-7 y^{2} x^{4}\nonumber\]

First, we rewrite the given expression using addition only:

\[5 x^{4} y^{2}+6 x^{3} y+\left(-7 y^{2} x^{4}\right)\nonumber\]

Here the terms are: \(5 x^{4} y^{2}, 6 x^{3} y,\) and \(-7 y^{2} x^{4}\). The terms are added to get the expression.

Two terms are either “unlike” or “like”.

Like Terms

Like terms are terms that have the same exponents on the same variables.

For example, in the above expression \(5 x^{4} y^{2}\) and \(-7 y^{2} x^{4}\) are like terms since \(y\) has the same exponent (2) and the \(x\) has the same exponent (4). On the other hand, \(5 x^{4} y^{2}\) and \(6 x^{3} y^{2}\) are unlike terms since \(x^{4}\) appears in the first term but \(x^{3}\) appears in the second.

Example 8.1

Consider the expression:

\(-2 x^{5} y^{2}-5 x^{4} y^{2}+6 x^{5} y^{2}\)

Solution

The like terms are: \(-2 x^{5} y^{2}\) and \(6 x^{5} y^{2}\).

Adding or Subtracting Like Terms

Adding or Subtracting Like Terms

We can only add or subtract like terms, and, we do so by adding or subtracting their coefficients.

Example 8.2

\(-2 x^{5} y^{2}+6 x^{5} y^{2}=4 x^{5} y^{2}\) so that our expression in the previous example can be simplified:

\[-2 x^{5} y^{2}-5 x^{4} y^{2}+6 x^{5} y^{2}=4 x^{5} y^{2}-5 x^{4} y^{2}\nonumber\]

Now, let’s add and subtract polynomials.

Example 8.3

Add or subtract the polynomials.

a) \(\begin{align*}
\left(3 x^{2}+5 x+6\right)+\left(4 x^{2}+3 x-8\right) &=3 x^{2}+5 x+6+4 x^{2}+3 x-8 \\
&=7 x^{2}+8 x-2
\end{align*}\)

b) \(\begin{align*}
\left(x^{4}-2 x^{3}+6 x\right)+\left(5 x^{3}+2 x^{2}+9 x+4\right) &=x^{4}-2 x^{3}+6 x+5 x^{3}+2 x^{2}+9 x+4 \\
&=x^{4}+3 x^{3}+2 x^{2}+15 x+4
\end{align*}\)

c) \(\begin{align*}
\left(2 a b^{2}-3 a^{2}-7 a b\right)+\left(-a^{2}-5 a^{2} b\right) &=2 a b^{2}-3 a^{2}-7 a b-a^{2}-5 a^{2} b \\
&=2 a b^{2}-4 a^{2}-7 a b-5 a^{2} b
\end{align*}\)

Note: Above, the only like terms that can be combined are \(-3 a^{2}\) and \(-a^{2}\) The remaining terms cannot be combined any further.

d) \(\begin{align*}
\left(3 x^{2}+5 x+6\right)-\left(4 x^{2}+3 x-8\right) &=3 x^{2}+5 x+6-4 x^{2}-3 x+8 \\
&=-x^{2}+2 x+14
\end{align*}\)

Note: Removing the parenthesis when subtracting changes the sign for all the terms of the polynomial that is being subtracted.

e) \(\begin{align*}
(5 m+3 n)-(7 m+2 n) &=5 m+3 n-7 m-2 n \\
&=-2 m+n
\end{align*}\)

f) Subtract \(4 a^{2}-5 a\) from \(6 a+4\).

\(\begin{align*}
(6 a+4)-\left(4 a^{2}-5 a\right) &=6 a+4-4 a^{2}+5 a \\
&=-4 a^{2}+11 a+4
\end{align*}\)

Note: Subtracting \(4 a^{2}-5 a\) from \(6 a+4\) requires to write \(6 a+4\) first, and then to subtract \(4 a^{2}-5 a\) from it. Reversing the order would yield a wrong answer!

g) Subtract \(-3 q+4 p q-2 p\) from \(-9 p\).

\(\begin{align*}
-9 p-(-3 q+4 p q-2 p) &=-9 p+3 q-4 p q+2 p \\
&=-7 p+3 q-4 p q
\end{align*}\)

Exit Problem

Simplify: \(\left(9 m^{2} n-15 m n^{2}\right)-\left(3 m n^{2}+2 m^{2} n\right)\)