Starlight Tools

Combine Like Terms Calculator — Expand and Simplify Expressions

Enter an expression, remove parentheses, collect matching terms, and see every step for free. This expression expander and distributive property calculator helps students check homework and teachers explain the algebra.

Example: 2(x + 3) + 3x − 4 → 5x + 2

Enter an expression or equation

Choose an operation

Expand only removes parentheses and keeps distributed terms separate. Combine only adds matching terms within each sum and keeps products containing sums in parentheses. Expand and combine does both.

2(x + 3) + x − 4 + 2x

Use single-letter variables such as x, y, a, b, numbers, + − * /, parentheses, powers 0–12, and at most one =. Implicit multiplication such as 3x and (x+1)(x−2) is accepted.

Input rules and scope

Coefficients may be integers, finite decimals, or fractions. Division is allowed only by a nonzero constant. Variables are case-sensitive Latin letters (a–z, A–Z); adjacent letters mean multiplication, so xy=yx. Roots, functions, negative powers, and variable denominators are unsupported. Multiplication and division are evaluated left to right: 1/2x means (1/2)x; use parentheses to clarify fractions. An equation is simplified on each side, not solved.

Try an example

Simplified result

Expanded and combined
5x + 2
Equivalent polynomial expression
Distributed terms
2x + 6 + x − 4 + 2x
Like-term groups
x: 2 + 1 + 2 = 5; constants: 6 − 4 = 2
Degree1
Terms in standard form2

Copy simplified answer always collects all terms. LaTeX and step exports follow the selected mode. Share links restore the last calculated expression and mode.

Expression expander: step-by-step working

How to expand expressions and collect like terms

1. Apply distribution

Multiply the factor outside parentheses by every term inside: a(b+c)=ab+ac. For two polynomials, multiply every term in the first by every term in the second.

2. Identify variable parts

Terms are alike only when their variable parts match exactly. 4x² and −x² are like terms; 4x² and 4x are not.

3. Add coefficients

Keep the shared variable part and add only its numerical coefficients: 4x²−x²=(4−1)x²=3x².

4. Write standard form

List nonzero terms by descending total degree (the sum of exponents), then by descending exponents of alphabetically ordered variables. Terms whose coefficients sum to zero disappear.

Distributive property calculator: the rule

a(b + c) = ab + ac

For example, 2(x+3)+x−4+2x expands to 2x+6+x−4+2x. Grouping gives (2+1+2)x+(6−4), so the simplified form is 5x+2.

Worked examples: expand, group, and simplify

Each line is an equivalent form. When there are no parentheses, the distributed form already matches the input.

Basic collection

Original
7x+3x-4
Distributed form
7x + 3x − 4
Grouped form
(7 + 3)x − 4
Coefficient calculation
7 + 3 = 10; constant = −4
Answer
10x − 4

Negative coefficient

Original
-3(x+2)+5x
Distributed form
−3x − 6 + 5x
Grouped form
(−3 + 5)x − 6
Coefficient calculation
−3 + 5 = 2; constant = −6
Answer
2x − 6

Subtraction before parentheses

Original
5x-(2x-4)
Distributed form
5x − 2x + 4
Grouped form
(5 − 2)x + 4
Coefficient calculation
5 − 2 = 3; −(−4) = +4
Answer
3x + 4

Two binomials: FOIL

Original
(x+2)(x-3)
Distributed form
x² − 3x + 2x − 6
Grouped form
x² + (−3 + 2)x − 6
Coefficient calculation
x²: 1; x: −3 + 2 = −1; constant: −6
Answer
x² − x − 6

Nested parentheses

Original
2(x+3(x-1))
Distributed form
2x + 6x − 6
Grouped form
(2 + 6)x − 6
Coefficient calculation
2 + 6 = 8; constant = −6
Answer
8x − 6

Perfect square

Original
(x+3)^2
Distributed form
x² + 3x + 3x + 9
Grouped form
x² + (3 + 3)x + 9
Coefficient calculation
x²: 1; x: 3 + 3 = 6; constant: 9
Answer
x² + 6x + 9

Fractional coefficients

Original
(1/2)x+(3/4)x-2
Distributed form
(1/2)x + (3/4)x − 2
Grouped form
(1/2 + 3/4)x − 2
Coefficient calculation
1/2 + 3/4 = 2/4 + 3/4 = 5/4
Answer
5/4·x − 2

Cancellation to zero

Original
2(x+1)-2x-2
Distributed form
2x + 2 − 2x − 2
Grouped form
(2 − 2)x + (2 − 2)
Coefficient calculation
2 − 2 = 0 for both groups
Answer
0

Multiple variables

Original
7x+2y+3x+4y
Distributed form
7x + 2y + 3x + 4y
Grouped form
(7 + 3)x + (2 + 4)y
Coefficient calculation
x: 7 + 3 = 10; y: 2 + 4 = 6
Answer
10x + 6y

Canonical variable order

Original
2xy+5yx
Distributed form
2xy + 5xy
Grouped form
(2 + 5)xy
Coefficient calculation
xy: 2 + 5 = 7
Answer
7xy

Different multivariable powers

Original
3a^2b-2ab^2
Distributed form
3a²b − 2ab²
Grouped form
3(a²b) − 2(ab²)
Coefficient calculation
a²b: 3; ab²: −2; these groups differ
Answer
3a²b − 2ab²

Both sides of an equation

Original
2(x+3)=x+x+6
Distributed form
2x + 6 = x + x + 6
Grouped form
2x + 6 = (1 + 1)x + 6
Coefficient calculation
Left x: 2; right x: 1 + 1 = 2; constants: 6
Answer
2x + 6 = 2x + 6

FOIL and special products

FOIL works for two binomials

Multiply First, Outer, Inner, Last: (a + b)(c + d) = ac + ad + bc + bd. For (x + 2)(x − 3), the four products are x·x, x·(−3), 2·x, and 2·(−3). This gives x² − 3x + 2x − 6 = x² − x − 6.

FOIL is a shortcut for exactly two binomials. For three-term or larger polynomials, use general distribution: every term in one factor multiplies every term in the other.

Square of a sum

(a + b)² = (a + b)(a + b) = a² + ab + ab + b² = a² + 2ab + b²

Square of a difference

(a − b)² = (a − b)(a − b) = a² − ab − ab + b² = a² − 2ab + b²

The last term is positive because (−b)(−b) = b². Do not omit the middle term.

Product of a sum and difference

(a + b)(a − b) = a² − ab + ab − b² = a² − b²

The two middle terms cancel. The calculator shows these products before collecting them.

Accuracy, editorial accountability, and privacy

Editorial owner: Starlight Tools (Starlight Robotics). Implementation and automated checks reviewed: . This identifies the site owner and technical verification date; it is not a claim of review by a named algebra educator.

Coefficients use reduced BigInt fractions. A finite decimal such as 0.125 is stored exactly as 1/8, so coefficient arithmetic introduces no binary floating-point rounding. The optional spot-check evaluates the original expression tree and the selected result separately at two assignments using exact fractions. Algebraic transformations establish equivalence; numeric samples alone are not proof.

Documented verification set

The page-specific regression suite (equation-expander.test.cjs) checks all worked examples above, all three modes, xy/yx matching, unary minus versus powers, nested parentheses, fractional division, zero cancellation, equation sides, malformed input, and expansion limits. Every recorded transformation is also evaluated against its original expression at representative values.

Supported input and limits

Use single Latin letters, exact integers or finite decimals, constant denominators, and integer powers 0–12. Limits are 400 normalized characters, total degree 12, 500 distributed terms, 18 digits per numeric literal, 220 digits per reduced numerator or denominator, and 1,600 expansion transformations per side. All modes use the same resource limits. The zero polynomial has no defined degree. A zeroth power is treated as 1 under the polynomial convention, including 0^0. The tool does not factor, solve equations, handle functions or roots, or cancel variable denominators.

Calculations run locally. Copy and download actions export only when selected. Shareable URLs store the expression and mode in the URL fragment; anyone with the link can read them. The site’s advertising and analytics policies remain separate from this calculator’s local processing.

Report an incorrect result

Describe the issue, then copy or download a report with your entered expression, mode, and last result attached. Nothing is sent automatically.

Email incorrect-result report

Opens your email app with the expression and report included, addressed to info@starlighttools.org (the contact published on our About page). Review the message before sending. If no email app is configured, copy the report into your webmail.

Combining like terms and expansion FAQs

What are like terms?

Like terms have identical variable parts, including every exponent. For example, 3a²b and −5a²b are like terms. Constants form their own group.

What is the difference between expanding and simplifying?

Expanding removes products of sums by distribution: 2(x+3)+x becomes 2x+6+x. Simplifying collects matching terms to give 3x+6. Use the mode controls to choose either operation or both.

How do you distribute a negative sign?

Treat the minus sign as multiplication by −1 and flip every sign: −(2x−3y+4) = −2x+3y−4.

Can x and x² be combined?

No. Their exponents differ, so they are different variable parts. The expression 3x+2x² has two groups.

Are xy and yx like terms?

Yes. Multiplication commutes, so xy=yx and 2xy+5yx=7xy. But x²y and xy² are different terms.

What happens when terms cancel to zero?

Their coefficient sum is zero, so the group disappears from the simplified answer. If every group cancels, the answer is 0. Expand only keeps the distributed terms separate.

Does combining like terms change the value?

No. Adding coefficients of identical variable parts preserves the value for every allowed assignment. The numeric spot-check illustrates this with two exact samples.

Does this calculator solve for x?

No. It expands and collects expressions. With one equals sign, it transforms the two sides independently without moving terms or finding a solution.

Can I enter fractions, decimals, and implicit multiplication?

Yes. For example, 0.5x+(3/4)x becomes 5/4·x. A divisor must be a nonzero constant. Write (1/2)x for half of x; x/y is unsupported. Products such as 3x, xy, and (x+1)(x−1) are accepted.

Does the calculator round coefficients?

No. Finite decimals and constant fractions use exact reduced rational arithmetic. Variable letters are case-sensitive, and function notation is unsupported.

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