Section 1.6
Combining Like Terms
Terms are expressions that are added or subtracted, as opposed to factors, which are expressions that are multiplied together. The expression $3x+7xy$ has two terms ($3x$ and $7xy$). It has no factors in the form in which it is written. However, the equivalent expression $x(3+7y)$ has two factors ($x$ and $3+7y$) but no like terms. Terms are considered "like terms" if they have the same variables raised to the same powers.
Examples:
a. To simplify the expression $5x^3y+7xy^3-2x^3y$, we note that $5x^3y$ and $-2x^3y$ are like terms. The $5x^3y$ and the $-2x^3y$ can be combined, while the $7xy^3$ must stay by itself.
Thus $5x^3y+7xy^3-2x^3y=3x^3y+7xy^3$.
b. To simplify the expression $(x+3)(x-2)+(x+7)(x-1)$, we first multiply out the binomials (although the expression has two terms to begin with, they are not "like terms").
$$(x+3)(x-2)+(x+7)(x-1)=x^2-2x+3x-6+x^2-x+7x-7$$
Now we see that there are two terms with $x^2$, there are four terms with $x$, and there are two numbers (constants). These three different kinds of terms can be combined.
\begin{align}
(x+3)(x-2)+(x+7)(x-1)&=x^2-2x+3x-6+x^2-x+7x-7\\
&=2x^2+7x-13
\end{align}
c. The expression $x+2xy+3y$ has three terms. However, they are of three different types---no like terms. This expression cannot be simplified.
Exercises:
Simplify each of the following expressions, if possible. If the expression cannot be simplified, say so.
\begin{array}{l l}
1. \;\;3{{x}^{2}}y-5x{{y}^{2}}+2{{x}^{2}}y+7x{{y}^{2}}\;\;& 2. \;\;2x+3{{y}^{2}}-7xy+5x-{{y}^{2}}\;\;\\
3. \;\;3{{x}^{2}}yz+2xyz-4xy(z+2xz)\;\;& 4. \;\;a(x+2b)-x(2a-3b)+b(a+x)\;\;\\
5. \;\;5+3(x+y)\;\; & 6. \;\;(x-4)(x+5)+(x-2)(x+2)\;\;\\
7. \;\;(x-2)(x+7)+(x-3)(x+1)\;\; & 8. \;\;(x+2)(x+7)-(x-5)(x+1)\;\;\\
9. \;\;(x+4)(x+3)-(x-2)(x-1)\;\; & 10. \;\;{{(x+4)}^{2}}+{{(x-4)}^{2}}\;\;\\
11. \;\;{{(x+3)}^{2}}-{{(x-3)}^{2}}\;\; & 12. \;\;{{(x+2)}^{2}}+(x+3)(x-7)\;\;\\
13. \;\;3(2x-1)+2(5x+3)-8(2x-3)\;\; & 14. \;\;{{(2x+3)}^{2}}-{{(x+1)}^{2}}\;\;\\
15. \;\;3(x-2)+7(x+1)-4(x-3)\;\; & 16. \;\;{{(3x-2)}^{2}}-{{(x+3)}^{2}}\;\;\\
\end{array}
17. Subtract $3{{x}^{2}}-2xy+{{x}^{2}}a$ from the sum of $3xy-2{{x}^{2}}-4a$ and $2{{x}^{2}}a+5{{x}^{2}}-5xy$.
18. Subtract $5x{{y}^{2}}K-2xy-3xK$ from the sum of $3xyK+2xK+7x{{y}^{2}}K$ and $xK-2x{{y}^{2}}K-2xy$.
19. The measurements of the barnyard are given in the diagram below. Determine a simplified expression for the number of feet of fencing required.
20. The measurements of the barnyard are given in the diagram below. Determine a simplified expression for the number of feet of fencing required.
21. Subtract the expression on the second line from that on the top.
$\begin{align}
& \text{a. }5x+3 \\
& \quad \underline{5x+8} \\
\end{align}$ $\begin{align}
& \text{b. }2{{x}^{2}}-7x \\
& \quad \underline{2{{x}^{2}}+5x} \\
\end{align}$ $\begin{align}
& \text{c. }-4{{x}^{3}}-3{{x}^{2}} \\
& \quad \underline{-4{{x}^{3}}-7{{x}^{2}}} \\
\end{align}$ $\begin{align}
& \text{d. }-3x-8 \\
& \quad \ \ \underline{-3x+5} \\
\end{align}$
22. Subtract the expression on the second line from that on the top.
$ \begin{align}
& \text{a. 7}x-3 \\
& \quad \underline{7x+5} \\
\end{align}$ $\begin{align}
& \text{b. 3}{{x}^{2}}-9x \\
& \quad \underline{3{{x}^{2}}-6x} \\
\end{align}$ $\begin{align}
& \text{c. }-2{{x}^{3}}+3{{x}^{2}} \\
& \quad \underline{-2{{x}^{3}}-7{{x}^{2}}} \\
\end{align}$ $\begin{align}
& \text{d. }-8x+3 \\
& \quad \ \ \underline{-\,8x-5} \\
\end{align}$
23. Identify each of the following as TRUE or FALSE.
\begin{array}{l l}
a. \;\;2+3x=5x\;\; & b. \;\;2x+3xy=5x+3y\;\;\\
c. \;\;{{2}^{5}}+{{2}^{5}}={{2}^{6}}\;\; & d. \;\;{{3}^{50}}+{{3}^{50}}+{{3}^{50}}={{3}^{51}}\;\;\\
e. \;\;3x(a+b)+5x(a+b)=8x(a+b)\;\;&\\
\end{array}
24. a. Add $-3(2x-3y)$ to $4(-x+2y)$.
b. Subtract $-2x+3y$ from $x-2y$.
25. Subtract the expression on the second line from that on the top.
$\begin{align}
& \text{a. }7{{x}^{2}}-3x-3 \\
& \quad \underline{7{{x}^{2}}+2x-5} \\
\end{align}$ $\begin{align}
& \text{b. }3{{x}^{3}}+4{{x}^{2}}-9x \\
& \quad \underline{3{{x}^{3}}-6{{x}^{2}}-8x} \\
\end{align}$ $\begin{align}
& \text{c. }-2{{x}^{4}}+2{{x}^{3}}-3{{x}^{2}} \\
& \quad \underline{-2{{x}^{4}}-7{{x}^{3}}+3{{x}^{2}}} \\
\end{align}$