Distribute
Expand
To multiply out the parts of an expression. Distributing is the
opposite of factoring.
| Example 1: |
3x(x + 8) = 3x·x + 3x·8
= 3x2 + 24x
|
| Example 2: |
(x + 2)(x + 5) = x(x + 5) + 2(x + 5)
= x·x + x·5 + 2·x + 2·5
= x2 + 7x + 10
|
| Example 3: |
(x – 3)4 = (x – 3)(x – 3)(x – 3)(x – 3)
= (x·x – x·3 – 3·x + 3·3)(x·x – x·3 – 3·x + 3·3)
= (x2 – 6x + 9)(x2 – 6x + 9)
= x2(x2 – 6x + 9) – 6x(x2 – 6x + 9) + 9(x2 – 6x + 9)
= x2·x2 – x2·6x + x2·9 – 6x·x2 + 6x·6x – 6x·9 + 9·x2 – 9·6x + 9·9
= x4 – 6x3 + 9x2 – 6x3 + 36x2 – 54x + 9x2 – 54x + 81
= x4 – 12x3 + 54x2 – 108x + 81
|
See
also
Distributing
rules, FOIL method, binomial theorem
Worked Example
Problem: Distribute and simplify: 4(2x + 5).
Step 1: Multiply the outside factor, 4, by the first term inside the parentheses, 2x.
4⋅2x=8x Step 2: Multiply the outside factor, 4, by the second term inside the parentheses, 5.
4⋅5=20 Step 3: Combine the two products using the same operation (addition) that appeared inside the parentheses.
4(2x+5)=8x+20 Answer: 4(2x + 5) = 8x + 20
Another Example
Problem: Distribute and simplify: -3(x² - 4x + 7).
Step 1: Multiply -3 by the first term, x².
−3⋅x2=−3x2 Step 2: Multiply -3 by the second term, -4x. A negative times a negative gives a positive.
−3⋅(−4x)=12x Step 3: Multiply -3 by the third term, 7.
−3⋅7=−21 Step 4: Write the full expanded expression.
−3(x2−4x+7)=−3x2+12x−21 Answer: -3(x² - 4x + 7) = -3x² + 12x - 21
Frequently Asked Questions
What does it mean to distribute in math?
Distributing means multiplying a factor outside the parentheses by every term inside the parentheses, one at a time, then writing the sum (or difference) of all those products. It relies on the distributive property of multiplication over addition: a(b + c) = ab + ac.
How do you distribute a negative sign?
Treat the negative sign as multiplying by -1. Multiply -1 by each term inside the parentheses, which flips the sign of every term. For example, -(3x - 2) becomes -3x + 2.
Distributing (Expanding) vs. Factoring
Distributing and factoring are inverse operations. Distributing removes parentheses by multiplying through—for example, turning 3(x + 4) into 3x + 12. Factoring reverses this by pulling out a common factor to rewrite 3x + 12 as 3(x + 4). If you can do one, you can check your work by doing the other.
Why It Matters
Distributing is one of the most frequently used algebraic skills. You need it to simplify expressions, solve equations, and multiply polynomials. Nearly every algebraic manipulation—from solving a basic linear equation to expanding binomials—depends on knowing how to distribute correctly.
Common Mistakes
Mistake: Multiplying only the first term inside the parentheses and forgetting the rest.
Correction: You must multiply the outside factor by every term inside the parentheses. For example, 5(x + 3) is 5x + 15, not 5x + 3.
Mistake: Dropping or flipping signs when distributing a negative number.
Correction: Track each sign carefully. When you distribute a negative, each term's sign changes. For instance, -2(x - 6) = -2x + 12, not -2x - 12.
