Multiplying variables follows two simple principles: multiply the numerical coefficients together, then combine the variables using exponent rules. Once you understand how these pieces work, you can handle everything from basic terms like 3x × 4x to more complex expressions with multiple variables and exponents.
The Basic Rule
Every variable term has two parts: a coefficient (the number in front) and one or more variables (the letters). When you multiply two terms, you handle each part separately. Multiply the coefficients as regular numbers, then combine the variables.
Take 6x × 5x as an example. First, multiply the coefficients: 6 × 5 = 30. Then combine the variables: x × x = x². The result is 30x². The x × x becomes x² because when you multiply the same variable, you add the exponents (x¹ × x¹ = x²). More on that in the next section.
When the variables are different, you simply write them next to each other. For 6x × 5y, you get 30xy. The x and y can’t be combined any further because they represent different quantities.
Adding Exponents With the Same Base
The product rule for exponents is the core mechanic behind multiplying variables. When two factors share the same base (the same variable letter), you add their exponents. So x³ × x⁴ = x⁷, because 3 + 4 = 7.
This works because exponents are shorthand for repeated multiplication. x³ means x × x × x, and x⁴ means x × x × x × x. Multiply them together and you have x written out seven times, which is x⁷.
A few examples to solidify the pattern:
- y² × y² = y⁴
- a⁵ × a = a⁶ (a lone variable has an invisible exponent of 1)
- z³ × z⁷ = z¹⁰
This rule only applies when the bases match. You cannot add the exponents of x³ × y² because x and y are different variables. That expression stays as x³y².
Multiplying Terms With Multiple Variables
Many algebra problems involve terms that contain more than one variable. The approach is the same: deal with the coefficients first, then handle each variable separately.
Consider 3x²y × 5xy³. Start with the coefficients: 3 × 5 = 15. Now handle the x variables: x² × x¹ = x³. Then the y variables: y¹ × y³ = y⁴. Put it all together and you get 15x³y⁴.
Another example: 2y² × 8z² = 16y²z². The coefficients multiply (2 × 8 = 16), and the variables stay separate because y and z are different bases.
Using the Distributive Property
When you multiply a term by an expression inside parentheses, you need the distributive property. This means multiplying the outside term by every term inside the parentheses, not just the first one.
For 2(x + 3), multiply 2 by x to get 2x, then multiply 2 by 3 to get 6. The result is 2x + 6.
The same logic applies when the outside term contains a variable. For x(x + 4), multiply x by x to get x², then x by 4 to get 4x. The result is x² + 4x.
With fractional coefficients, the process is identical. To distribute ½ across (6x + 8), multiply ½ × 6x = 3x and ½ × 8 = 4, giving you 3x + 4. The key is making sure you distribute to every term inside the parentheses. Skipping a term, especially when there are three or more, is one of the most common errors in algebra.
Multiplying Two Binomials
When both factors contain multiple terms, like (x + 2)(x + 5), you need to multiply each term in the first set of parentheses by each term in the second. This is sometimes called FOIL, which stands for First, Outer, Inner, Last.
- First: x × x = x²
- Outer: x × 5 = 5x
- Inner: 2 × x = 2x
- Last: 2 × 5 = 10
Add the results: x² + 5x + 2x + 10. Then combine the like terms (5x and 2x) to get x² + 7x + 10.
FOIL is really just the distributive property applied twice. It works for any pair of two-term expressions. For larger expressions with three or more terms on each side, you still multiply every term by every other term, then combine like terms at the end.
Handling Negative Signs
Negative signs follow the standard rules of multiplication. A positive times a negative gives a negative result. A negative times a negative gives a positive result.
For example, (-3x)(4x) = -12x². The coefficients multiply (-3 × 4 = -12) and the variables combine (x × x = x²). And (-2x)(-5x) = 10x², because negative times negative is positive.
Where negative signs get tricky is distribution. In an expression like -2(x – 3), you need to distribute the -2 to both terms: -2 × x = -2x, and -2 × (-3) = +6. The result is -2x + 6. Forgetting to carry the negative sign through to every term is a frequent source of errors, especially when the expression inside the parentheses already contains subtraction.
Quick Reference for Multiplying Variables
- Same variable: add the exponents. x² × x³ = x⁵
- Different variables: write them side by side. x² × y³ = x²y³
- Coefficients: multiply them as regular numbers. 4x × 7x = 28x²
- Parentheses: distribute to every term inside. 3(x + 2) = 3x + 6
- Negative signs: follow sign rules and distribute carefully. -2(x – 4) = -2x + 8