Distributive Property Calculator
This Distributive Property Calculator opens the brackets in an algebraic expression by multiplying the term outside each bracket by every term inside it, and shows the working at every step. The sign in front of a bracket travels with the term outside it, so a minus there flips the sign of everything it reaches. Once no brackets are left the expression is a plain sum, and any terms sharing a variable part are joined. Following the same pattern every time is what makes expanding easy to picture, check, and repeat on your own with any expression.
Step-by-step method
- Set up the problem.
- Write the formula.
- Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
- Do a combining like terms on what is left.
Formula:
See Example 1 Hide Example 1
Problem
Approach
Step-by-step method
- Set up the problem.
- Write the formula.
- Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
- Do a combining like terms on what is left.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(3\left(x + 4\right)\), with \(1\) bracket to open.
Step 2
Step 2 - Write the formula.
In this problem: Every bracket follows \(a\left(b + c\right) = ab + ac\), with the sign in front of a term travelling with it. Here \(a = 3\), \(b = x\) and \(c = 4\).
Step 3
Step 3 - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The \(3\) in front of \(\left(x + 4\right)\) multiplies every term inside it: \(3 \cdot x = 3x\) and \(3 \cdot 4 = 12\). No two terms in \(3x + 12\) share the same variable part, so there is nothing left to join and this is the answer.
Final Answer
See Example 2 Hide Example 2
Problem
Approach
Step-by-step method
- Set up the problem.
- Write the formula.
- Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
- Do a combining like terms on what is left.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(3\left(x + 4\right) - 2\left(y - 5\right)\), with \(2\) brackets to open.
Step 2
Step 2 - Write the formula.
In this problem: Every bracket follows \(a\left(b + c\right) = ab + ac\), with the sign in front of a term travelling with it. In bracket \(1\), \(a = 3\), \(b = x\) and \(c = 4\). In bracket \(2\), \(a = -2\), \(b = y\) and \(c = -5\).
Step 3
Step 3 - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The brackets here are \(3\left(x + 4\right)\) and \(-2\left(y - 5\right)\). They do not feed into each other, so the order does not change the answer and they are opened left to right.
Step 3a
Step 3a - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The \(3\) in front of \(\left(x + 4\right)\) multiplies every term inside it: \(3 \cdot x = 3x\) and \(3 \cdot 4 = 12\).
Step 3b
Step 3b - Brackets come first in the order of operations, so open them before anything else. The terms inside cannot be added together, so instead take one bracket at a time and multiply the term outside it by every term inside, keeping the sign that sits in front of each one.
In this problem: The \(-2\) in front of \(\left(y - 5\right)\) multiplies every term inside it: \(\left(-2\right) \cdot y = -2y\) and \(\left(-2\right) \cdot \left(-5\right) = 10\).
Step 4
Step 4 - Do a combining like terms on what is left.
In this problem: The brackets are gone, so \(3x + 12 - 2y + 10\) is now a plain sum. Joining the terms that match gives \(3x - 2y + 22\), worked through below.
Problem
Approach
Step-by-step method
- Set up the problem.
- List the terms in groups, putting every term with the same variable part into one group and all the plain numbers into a group of their own.
- Rewrite the expression with each group's terms next to each other, keeping every sign attached to the term in front of it.
- Take one group at a time and add or subtract the coefficients inside it, leaving the variable part unchanged.
Step 1
Step 1 - Set up the problem.
In this problem: The expression is \(3x + 12 - 2y + 10\), which is a sum of \(4\) terms.
Step 2
Step 2 - List the terms in groups, putting every term with the same variable part into one group and all the plain numbers into a group of their own.
In this problem: Reading the terms left to right, the variable parts are \(x\), none, \(y\), none. Matching them up gives \(3\) groups: the \(x\) terms with \(1\) of them, the \(y\) terms with \(1\) of them, the plain numbers with \(2\) of them.
Step 3
Step 3 - Rewrite the expression with each group's terms next to each other, keeping every sign attached to the term in front of it.
In this problem: The groups are not side by side yet, so the terms are written out again in the order the \(x\) terms, the \(y\) terms and the plain numbers, each one keeping the sign in front of it, which gives \(3x - 2y + 12 + 10\).
Step 4a
Step 4a - Take one group at a time and add or subtract the coefficients inside it, leaving the variable part unchanged.
In this problem: The plain numbers are \(12 + 10\). Adding them gives \(12 + 10 = 22\). The remaining \(3x\) and \(2y\) have nothing to pair with, so they come down unchanged.
Final Answer
Final Answer
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Example Problems
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