Distributive Property Calculator

What is the Distributive Property?

The distributive property of multiplication over addition is one of the most frequently used algebraic properties. It states that multiplying a number by a sum is equivalent to multiplying each addend separately and then adding the products together.

In algebraic notation, the fundamental distributive law is written as:

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

Similarly, for subtraction:

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

The FOIL Method for Binomials

When multiplying two binomial expressions, the distributive property is applied twice. This technique is commonly taught through the acronym FOIL:

$$(a_1 x + b_1)(a_2 x + b_2) = \underbrace{a_1 a_2 x^2}_{\text{First}} + \underbrace{a_1 b_2 x}_{\text{Outer}} + \underbrace{b_1 a_2 x}_{\text{Inner}} + \underbrace{b_1 b_2}_{\text{Last}}$$

First (F): Multiply the first terms in each set of parentheses: $(a_1 x)(a_2 x) = a_1 a_2 x^2$.
Outer (O): Multiply the outermost terms: $(a_1 x)(b_2) = a_1 b_2 x$.
Inner (I): Multiply the innermost terms: $(b_1)(a_2 x) = b_1 a_2 x$.
Last (L): Multiply the last terms in each bracket: $(b_1)(b_2) = b_1 b_2$.
Combine Like Terms: Add the Outer and Inner linear terms together: $(a_1 b_2 + b_1 a_2)x$.

Expanding Polynomials (Binomial $\times$ Trinomial)

When expanding larger expressions, such as $(a_1 x + b_1)(a_2 x^2 + b_2 x + c_2)$, each term of the first polynomial must be multiplied by every term in the second polynomial:

$$(a_1 x + b_1)(a_2 x^2 + b_2 x + c_2) = a_1 x(a_2 x^2 + b_2 x + c_2) + b_1(a_2 x^2 + b_2 x + c_2)$$

After distributing, terms of the same degree are grouped and combined into standard polynomial form.

Reverse Distribution: Factoring Out the GCF

The distributive property operates in both directions. Going in reverse is known as factoring out the greatest common factor (GCF):

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

For example, given $12x + 18$, the greatest common factor of $12$ and $18$ is $6$. Factoring $6$ out yields $6(2x + 3)$.

Related Algebraic Calculators

Continue simplifying and solving algebraic expressions with our dedicated tools: Quadratic Calculator, Discriminant Calculator, Linear Equation Solver, and Direct Variation Calculator.

Frequently Asked Questions

Why is the distributive property important?

The distributive property allows you to remove parentheses from algebraic expressions, combine like terms, solve linear and quadratic equations, and factor complex polynomials.

Does the distributive property apply to division?

Yes, division distributes over addition in the numerator: $\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$. However, addition does not distribute across a sum in the denominator: $\frac{c}{a + b} \neq \frac{c}{a} + \frac{c}{b}$.

Can the FOIL method be used for trinomials?

The FOIL acronym specifically applies to multiplying two binomials (each with 2 terms). For polynomials with 3 or more terms, use the general distributive rule where every term in the first polynomial multiplies every term in the second.

How do negative signs affect distribution?

When distributing a negative term, the sign of every term inside the parentheses reverses: $-a(b - c) = -ab + ac$.

Report