Distributive Property a(b+c)

Apply distributive property.

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Distributive Property Calculator: Mastering Algebra Expansion

The **distributive property** is one of the most fundamental rules in algebra. It allows you to multiply a single term by a sum or difference, "distributing" the multiplication across each term inside the parentheses. Our calculator makes this process instant.

What Is the Distributive Property?

The formal definition states:

$$a(b + c) = ab + ac$$

In words: "Multiply the outside term by each term inside the parentheses."

Why "Distributive"?

The word "distribute" means to hand out or spread. When you multiply $a$ by $(b + c)$, you're "handing out" $a$ to both $b$ and $c$.

Examples

Basic Example

$$3(x + 4) = 3 \cdot x + 3 \cdot 4 = 3x + 12$$

With Subtraction

$$5(2x - 7) = 5 \cdot 2x - 5 \cdot 7 = 10x - 35$$

Note: The negative sign stays with the term.

Negative Outside Term

$$-2(3 + x) = -2 \cdot 3 + (-2) \cdot x = -6 - 2x$$

Reverse: Factoring

The distributive property works in reverse too, called **factoring**:

$$6x + 9 = 3(2x + 3)$$

Here, we "factor out" the common factor (3).

FOIL Method (Special Case)

When multiplying two binomials, the distributive property extends to the famous **FOIL** method:

$$(a + b)(c + d) = ac + ad + bc + bd$$

First, Outer, Inner, Last

Example: $(x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6$

Real-World Application

Suppose you buy 3 combo meals, each with a burger ($b$) and fries ($f$). The total cost is:

$$3(b + f) = 3b + 3f$$

This is the distributive property in action!

Common Mistakes

  • Forgetting to distribute the negative: $-2(x - 3) \neq -2x - 3$. Correct: $-2x + 6$
  • Only multiplying the first term: $4(x + 5) \neq 4x + 5$. Correct: $4x + 20$
  • Sign errors: Watch out for double negatives!

Why It Matters

The distributive property is essential for:

  • Simplifying algebraic expressions
  • Solving equations
  • Factoring polynomials
  • Understanding polynomial multiplication

Conclusion

Master the distributive property, and algebra becomes significantly easier. Use our **Distributive Property Calculator** to verify your work and build confidence.