This page will show examples of adding and subtracting like terms and unlike terms, as well as multiplying algebraic expressions.
There are times when you can perform these operations on algebraic expressions, and also times when you can’t.
Adding and Subtracting Like Terms:
With algebraic expressions, addition and subtraction can only be done when we are adding and subtracting like terms.
Which were touched on in the Algebra 1 basics page.
But to recap, like terms are where the variable and the variable exponent are the same.
For example, 4x^2 and 5x^2 are like terms.
But 4x^2 and 5x^3 are NOT like terms, as the exponents are different.
Also 4x^2 and 5y^2 are NOT like terms, as the variables are different.
Examples
(1.1)
2x^4 + x^4 + 3x^3
= 3x^4 + 3x^3
(1.2)
4y \space - \space y^2 + 2a \space - \space a
= 4y \space - \space y^2 + a
Multiplying Algebraic Expressions Examples
Now unlike before where we had to only focus on subtracting and adding like terms in an expressions, it’s the case that unlike terms can be multiplied together when dealing with multiplying algebraic expressions examples.
With multiplying algebraic expressions, the exponent laws which can be seen on this page, are the important points to remember when dealing with such situations.
(2.1)
2(x \space - \space 2) = 2x \space - \space 4
(2.2)
x^{2}(a \space - \space 4x^{3}) = x^{2}a \space - \space 4x^5
(2.3)
(y + 1)(y^2 \space - \space x) = y^3 \space - \space yx + y^2 \space - \space x
Multiplying Algebraic Expressions,
Vertical Multiplication
Another way of performing multiplication on algebraic expressions is with what is known as the ‘vertical multiplication’ method.
The approach is similar to multiplying numbers together, and this method can be particularly useful when multiplying larger expressions together in Algebra.
Example
Let’s look at the vertical multiplication steps with the following example.
(x + 2) \times (2x + 3).
1)
First step is to line the relevant expressions up on top of each other vertically.
\begin{array}{r} {x+2}\space\space\\ \times \space\space\space\space\space{2x+3}\space\space\\ \hline \space\space \end{array}
2)
Multiply the lower number on the right with both the top terms, and bring the results down.
[/katex]\begin{array}{r} {x+2}\space\space\\ \times \space\space\space\space\space{2x+3}\space\space\\ \hline 3x+6\space\space \end{array}[/katex]
3)
Now multiply the lower number on the left with both the top terms, and again bring the results down to the relevant columns.
\begin{array}{r} \space\space\space\space{x+2}\space\space\\ \times \space\space\space\space\space\space\space\space\space{2x+3}\space\space\\ \hline \space\space\space\space3x+6\space\space\\ 2x^{2}+4x\space\space\space\space\space\space\space\space\space \end{array}
4)
Then we draw a horizontal line below the multiplication results, at which point now adding them together will give us the full result of the multiplication of the expressions.
\begin{array}{r} \space\space\space\space{x+2}\space\space\\ \times \space\space\space\space\space\space\space\space\space{2x+3}\space\space\\ \hline \space\space\space\space3x+6\space\space\\ 2x^{2}+4x\space\space\space\space\space\space\space\space\space\\ \hline 2x^{{\tiny{2}}}+7x+6\space\space \end{array}
=> (x + 2) \times (2x + 3) = 2x^{2} + 7x + 6