Simplify Calculator

About the Simplify Calculator

Simplifying is the everyday housekeeping of algebra. Before you can solve, graph or compare expressions, you usually need them in a clean standard form: brackets multiplied out, like terms collected and fractions reduced. Doing this by hand is mostly careful bookkeeping, and a single dropped sign spoils the result.

This calculator simplifies polynomial expressions in one or several variables, such as $3(x + 2) - 2(x - 1)$ or $(a + b)^2 - (a - b)^2$ , numeric expressions with fractions, and rational expressions in one variable, such as $\frac{x^2 - 9}{x^2 + 3x}$ , where it factors and cancels common factors and states the values that must be excluded.

What does it mean to simplify an expression?

Two expressions are equivalent when they give the same value for every allowed input. Simplifying replaces an expression with an equivalent one that is shorter or more standard. For polynomials, the standard form has no brackets, one term for each combination of powers, and terms written in descending degree.

For rational expressions, simplest form means numerator and denominator share no common polynomial factor. Cancelling a factor such as $x + 3$ is only valid where $x + 3 \ne 0$ , so a complete answer records that restriction.

How to use the Simplify Calculator

Type the expression, for example 3(x + 2) - 2(x - 1) or (x^2 - 9)/(x^2 + 3x).
Use ^ for powers and * or juxtaposition for multiplication: 2x, 2*x and 2(x + 1) all work.
Press Simplify and follow the steps: distribute, expand powers, combine like terms, and factor and cancel for fractions.
Read the final simplified form and any excluded values listed with it.

Formulas

Distributive law

a(b + c) = ab + ac

Square of a binomial

(a + b)^2 = a^2 + 2ab + b^2

Combining like terms

px^k + qx^k = (p + q)x^k

Cancelling a common factor

\frac{AC}{BC} = \frac{A}{B}, \quad C \ne 0

Worked example: 3(x + 2) − 2(x − 1)

The example simplifies $3(x + 2) - 2(x - 1)$ . The engine distributes each coefficient, taking care that $-2$ multiplies both terms in the second bracket, which turns $-1$ into $+2$ .

That gives $3x + 6 - 2x + 2$ . Collecting the $x$ terms and the constants separately leaves $x + 8$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

x + 8

How we read your problem

Simplify the polynomial expression in x.

3 \left( x+2\right)-2 \left( x-1\right)

Step-by-step solution

Read the expression
$3 \left( x+2\right)-2 \left( x-1\right)$
Distribute (multiply out every product)
Each term in one factor multiplies each term in the other: a(b + c) = ab + ac.
Result after multiplying out
$3x + 6 - 2x + 2$
Combine like terms
x: 3 − 2 = 1; constants: 6 + 2 = 8. Like terms have identical variables and exponents, so only their coefficients are added.
Write the result in standard form
Terms are ordered by descending total degree.
$x + 8$

Formulas used

Distributive property
$a(b + c) = ab + ac$
Square of a sum
$(a + b)^2 = a^2 + 2ab + b^2$
Like terms
$m\,x^k + n\,x^k = (m + n)\,x^k$

After expanding and combining like terms the expression is x + 8, a polynomial of degree 1 with 2 terms.

Verification

Spot check at x = 0.73: original 8.73, simplified 8.73 ✓

Detailed explanation

Like terms

Like terms have exactly the same variables raised to exactly the same powers; only their numerical coefficients may differ. So $5x^2y$ and $-2x^2y$ are like terms and combine to $3x^2y$ , but $x^2y$ and $xy^2$ are not like terms and must stay separate.

The calculator stores each term as a coefficient attached to a list of exponents. Adding two expressions then means adding the coefficients that belong to the same list, which is exactly what combining like terms means. Because coefficients are exact fractions, results such as $\tfrac{1}{3}x + \tfrac{1}{6}x = \tfrac{1}{2}x$ come out exact.

Expanding before combining

Brackets hide like terms. In $(a + b)^2 - (a - b)^2$ nothing can be combined until both squares are expanded: $a^2 + 2ab + b^2 - (a^2 - 2ab + b^2)$ . Then the $a^2$ and $b^2$ terms cancel, leaving $4ab$ . The calculator always expands every product and power first, then collects.

Subtracting a bracket is a common source of errors. The minus sign applies to every term inside, so $-(a^2 - 2ab + b^2) = -a^2 + 2ab - b^2$ .

Rational expressions

To simplify a fraction of polynomials, factor the numerator and denominator completely and cancel factors that appear in both. For $\frac{x^2 - 9}{x^2 + 3x} = \frac{(x - 3)(x + 3)}{x(x + 3)}$ the common factor $x + 3$ cancels, giving $\frac{x - 3}{x}$ with $x \ne 0$ and $x \ne -3$ .

You can cancel only factors, never individual terms. Cancelling the $x^2$ in $\frac{x^2 - 9}{x^2 + 3x}$ would be wrong, because $x^2$ is a term of a sum, not a factor of the whole numerator.

Numeric expressions and the order of operations

When an expression contains only numbers, simplifying means evaluating it exactly. The calculator follows the usual order: brackets first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. Fractions are kept as fractions throughout, so $\tfrac34 + \tfrac56 \cdot 2$ becomes $\tfrac34 + \tfrac53 = \tfrac{29}{12}$ rather than a rounded decimal.

Exact arithmetic matters because rounding early can change a later result. If you need a decimal for a practical answer, convert the final exact value once, at the very end.

Common mistakes

Distributing a negative sign to only the first term inside a bracket.
Writing $(a + b)^2$ as $a^2 + b^2$ and forgetting the middle term $2ab$ .
Combining unlike terms, such as adding $x^2$ and $x$ to get $2x^3$ .
Cancelling terms instead of factors in a fraction.
Dropping the excluded values after cancelling a factor.

Tips

Rewrite subtraction of a bracket as adding $-1$ times the bracket before distributing.
Underline like terms in the same colour before collecting them.
Check a simplification by substituting a simple value such as $x = 2$ into both forms.
Factor completely before cancelling anything in a rational expression.
Write the result in descending powers so it is easy to compare with other answers.

Frequently asked questions

What are like terms?

Terms with the same variables raised to the same powers, such as 4xy and minus 7xy. Only their coefficients differ, so they can be added into a single term.

Does simplifying change the value of an expression?

No. A simplified expression is equivalent to the original for every allowed input. The only difference is that cancelled factors create excluded values that must be stated.

Can the calculator simplify expressions with several variables?

Yes, polynomial expressions in any number of variables are expanded and their like terms combined. Rational expressions are simplified in one variable.

Why does my simplified fraction have restrictions?

When a factor is cancelled from top and bottom, the original expression was undefined where that factor is zero. The simplified form must keep that restriction to remain equivalent.

Is simplifying the same as solving?

No. Simplifying rewrites an expression; solving finds the values that make an equation true. Simplify both sides first, then use the equation solver if you need the solutions.

What is standard form for a polynomial?

All brackets multiplied out, one term for each combination of powers, and the terms written from the highest degree to the lowest, with the constant last.

Can I simplify numeric expressions too?

Yes. Expressions such as 3/4 + 5/6 times 2 are evaluated exactly with the order of operations and reduced to lowest terms.

Simplify your expression

Paste any expression above to see it expanded, collected and reduced step by step.

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