Lesson 3: Distribution, Subtraction, and Factoring

Property

If $a, b, c$ are real numbers, then
$a(b+c) = ab+ac$
$(b+c)a = ba+ca$
$a(b-c) = ab-ac$
$(b-c)a = ba-ca$

Examples

• To simplify $7(x+2)$, distribute the $7$: $7 \cdot x + 7 \cdot 2 = 7x + 14$.
• To simplify $-5(y-4)$, distribute the $-5$: $(-5) \cdot y - (-5) \cdot 4 = -5y + 20$.
• To simplify $9 - 3(x+1)$, first distribute the $-3$: $9 - 3x - 3$. Then combine like terms to get $6 - 3x$.

Explanation

The distributive property lets you 'distribute' or 'pass out' the number outside the parentheses to every term inside. It's the key to removing parentheses and simplifying expressions in algebra. Think of it as sharing the multiplication.

Property

To combine like terms, first identify the terms that have the same variables and exponents.
Next, rearrange the expression so the like terms are grouped together.
Finally, add or subtract the coefficients of the like terms to simplify the expression.

Examples

• To simplify $7a + 4b + 2a$, we identify $7a$ and $2a$ as like terms and combine them to get $9a + 4b$.
• In the expression $5x^2 + 9 + 2x^2 - 4$, we combine the $x^2$ terms to get $7x^2$ and the constants to get $5$. The result is $7x^2 + 5$.
• To simplify $10y + 3y^2 + 2y + 6y^2$, we combine the $y^2$ terms to get $9y^2$ and the $y$ terms to get $12y$. The simplified expression is $9y^2 + 12y$.

Explanation

Combining like terms simplifies an expression by grouping similar items. Just as you would group 3 apples and 4 apples to get 7 apples, you combine terms like $3x$ and $4x$ to get $7x$.

Property

When subtracting expressions, distribute the negative sign to ALL terms inside parentheses: $(a + b) - (c + d) = a + b - c - d$, not $a + b - c + d$.

Examples