Simplify Expression
Enter a polynomial expression in x. The calculator distributes every product, expands integer-power binomials and collects all like terms into a single polynomial in standard form. Useful when an answer needs to come out as a single expression rather than a sum of products.
A canonical form for comparison
An algebraic expression can be written many ways and mean the same thing. A sum of products, a product of sums, a power left unexpanded — all describe one function, and none is inherently more correct than the others. Standard form is simply the one everybody can compare against.
This calculator produces it. Every product is distributed, every integer power expanded, and every like term collected into a single polynomial with the powers in order.
The practical use is settling whether two expressions are the same. Written differently they can look unrelated, and expanding both to standard form makes the question decidable by inspection rather than by argument.
It is also the form most later operations expect. Reading off a degree, identifying a leading coefficient, or feeding an expression into differentiation all assume the terms have been collected, and an unexpanded product hides all three.
How to use this calculator
- Type the expression in x Brackets, plus and minus, multiplication and integer powers with ^. Nested brackets are fine and are expanded from the inside out.
- Keep it polynomial Division by an expression, and any trigonometric, logarithmic or exponential term, takes the input outside what this tool handles and is rejected with a message saying so.
- Read the standard form One polynomial, terms in descending order of power, with like terms already combined and zero terms omitted.
How the expression is reduced
The input is parsed into an expression tree and then expanded into a map from powers to coefficients. That representation is what does the work: once an expression is a coefficient list, distributing and collecting have already happened, because there is nowhere for an unexpanded product to hide.
The expansion is recursive. A product of two sub-expressions expands each side first and then multiplies the two coefficient lists together, which combines every term of one with every term of the other. An integer power is handled as repeated multiplication of the base by itself.
Collecting like terms is not a separate pass. Two terms of the same power land on the same key in the map and their coefficients are added as they arrive, so no expression ever exists in a state where like terms are present but uncombined.
If any part of the input cannot be reduced to a coefficient list — a division by something containing x, or a named function — the expansion fails and the tool reports that rather than returning a partially simplified result. Nothing is guessed at.
What each input means
- expr Expression — form field “Expression in x”
- A polynomial expression in x. It may contain brackets, integer powers and nested products, and need not be in any particular arrangement.
- result Standard form
- A single polynomial with terms in descending order of power and like terms combined. Zero coefficients are omitted rather than shown.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A product added to another term
Two brackets multiplied together, with a separate squared term added on. Both an expansion and a collection are needed.
Inputs Expression in x = (x + 1)(x - 2) + x^2
- Expression (x + 1)(x - 2) + x^2
- Expand all products Distribute every product and raise each power to its expanded form.
- Collect like terms Sum coefficients of matching powers of x.
- Simplified form 2x² − x − 2
Result 2x² − x − 2
The product expands to a quadratic, and the extra squared term then merges with its leading term. The result is a quadratic whose leading coefficient is 2 rather than 1 — which is invisible until the collection happens.
Nothing here could be read off the input directly. The degree is apparent, but the leading coefficient and the constant both require the expansion to be carried out.
Collecting without expanding
A sum of like terms and a constant, with no brackets at all. Only the collection step does anything.
Inputs Expression in x = 3x + 2x - 5
- Expression 3x + 2x - 5
- Expand all products Distribute every product and raise each power to its expanded form.
- Collect like terms Sum coefficients of matching powers of x.
- Simplified form 5x − 5
Result 5x − 5
The two x terms combine into one, and the constant is left alone since nothing matches it. This is the simplest case the tool handles and the one where standard form is closest to the input.
Even here the terms are reordered into descending powers. Standard form fixes the order as well as the grouping, which is part of what makes it canonical.
Squaring a binomial
A bracket raised to the second power. The power is expanded as repeated multiplication rather than distributed term by term.
Inputs Expression in x = (x - 1)^2
- Expression (x - 1)^2
- Expand all products Distribute every product and raise each power to its expanded form.
- Collect like terms Sum coefficients of matching powers of x.
- Simplified form x² − 2x + 1
Result x² − 2x + 1
The answer has three terms, not two. Squaring a difference produces a cross term, and its absence is the classic error this expansion makes visible.
The middle term is negative because one of the two factors is. Expanding the bracket against itself is what produces that sign correctly, whereas squaring each term separately loses it entirely.
Reading the result
Standard form is unique
Two expressions describing the same function always expand to the same polynomial. That is what makes expansion a test of equivalence rather than merely a tidying step.
Simpler is not always shorter
Expanding a product usually produces more terms than it started with. Standard form is canonical rather than compact, and a factored expression is often the more useful one to work with.
A missing term means a zero coefficient
A gap in the powers is not an omission — it means that power's coefficient came out as zero after collection. Reading the degrees present is how a cancellation shows itself.
When you would use this
Checking whether two answers agree
A differently arranged but correct answer is common in algebra. Expanding both to standard form and comparing term by term settles it definitively.
Preparing an expression for another operation
Reading off a degree or a leading coefficient, or feeding a polynomial into the derivative or factoring tools, all assume the collected form. Expanding first removes any ambiguity about what the expression actually is.
Assumptions and limitations
What this calculator assumes
- The expression is a polynomial in x, with non-negative whole-number powers.
- All products and integer powers are expanded fully; nothing is left in factored form.
- Like terms are combined as the expansion proceeds rather than in a separate pass.
- An input that cannot be reduced to a coefficient list is rejected rather than partially simplified.
Where it stops being the right tool
- No division by an expression, and no trigonometric, logarithmic or exponential terms.
- One variable: expressions in two or more variables are outside the coefficient-list representation used here.
- No factoring — this tool only goes in the expanding direction, and the factoring calculator does the reverse.
- Fractional and negative exponents are not supported.
Common mistakes
Squaring a bracket term by term
Why it happens. Squaring distributes over a product, so applying it to a sum feels equally valid. It loses the cross term, which is usually the largest of the three.
How to avoid it. Write the bracket twice and expand as a product. The squared example above shows the term that goes missing.
Dropping a sign when subtracting a bracket
Why it happens. A minus in front of a bracket has to reach every term inside it, and the terms after the first are easy to leave untouched.
How to avoid it. Distribute the minus explicitly before combining. Comparing your constant term against the tool's usually reveals it immediately.
Expecting the result to be shorter
Why it happens. Simplify suggests compression, and a factored expression genuinely is more compact than its expansion.
How to avoid it. Read the output as standard form rather than as a shorter form. If compactness is what you want, factoring is the operation to reach for.
Frequently asked questions
What inputs are accepted?
Any polynomial expression in x built from plus, minus, multiplication, brackets and integer powers. Division by an expression and trigonometric, logarithmic or exponential terms are outside what this handles, and are rejected with an explanation.
Does it factor the result?
No — it expands and collects, which is the opposite direction. The factoring calculator takes a polynomial and produces its factored form, and running one after the other returns you to where you started.
What if my expression isn't a polynomial?
The tool reports a clear error rather than partially simplifying. For general expressions in x, the symbolic derivative and integral calculators accept a much wider range of functions.
Why is the answer longer than what I typed?
Because expanding a product produces more terms than the factored form contains. Standard form is canonical rather than compact — its value is that every equivalent expression reduces to exactly the same thing.