How do I combine like terms?

How it works

The page reads your expression and writes it as a sum of terms. Each term is an exact fraction (the coefficient) times a product of factors, each factor to a whole-number power. A factor is a letter, or a piece kept whole:

• a function of something, such as sin(2x) or ln(x + 1) (its inside is simplified the same way first);
• π or e;
• a power whose exponent is not a whole number, such as x^(1/2) or e^x;
• a sum under a fraction bar, such as (x + 1) in 3/(x + 1) (a sum to a negative whole power down to −20 goes under the bar the same way);
• a sum to any other power, such as (x + 1)^25, kept whole.

The rules:

• Brackets: products are multiplied out (every term times every term); a sum to a whole-number power from 0 to 20 is multiplied out; a single term to any whole power up to ±100 is kept as one term; any power 0 is 1, so x^0 is the number 1 and adds to the other numbers.
• Division: dividing by a number or by a single term (such as 4y²) divides the coefficient and subtracts powers; dividing by a sum keeps that sum whole under the bar. A sum on top is not cancelled against a sum underneath: (x + y)²/(x + y) stays as x²/(x + y) + 2x y/(x + y) + y²/(x + y).
• Like terms: terms with the same factors to the same powers are added; a term whose coefficient becomes 0 is dropped.
• Numbers are exact: 0.1 is 1/10, 2.5 is 5/2. Dividing by 0 gives a message.

Check. The written answer is read back and evaluated at 15 test points (each letter set to 3 sin(3i + j) for point i = 1 to 15, where j counts the letters a, b, c, d, f, … from 0); at each point both must be real and agree to 10⁻⁹ × max(1, |value|), or neither may be real.

What you can type

• Letters: any lowercase letter except e (Euler’s number), and θ. pi (or π) is π.
• Operations: + − * / and ^; brackets group; numbers and letters side by side multiply (3x y means 3 · x · y).
• Functions: sqrt, cbrt, ln (and log, written ln), log10, exp, abs, sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh and their inverses.

How answers are written

• Terms whose factors are all letters come first: highest total power first, then by the powers of the letters in alphabetical order (x²y before xy² before y²). Terms with other pieces come next; the number with no letter comes last. 0 when everything cancels.
• In a term: the numerator of the coefficient (left out when it is 1), then the letters in alphabetical order, then the other pieces; powers above 1 as ^ (x^2); a negative power or a coefficient’s denominator goes under a fraction bar: 5x/6, 3/x, 7x/(4y^2).
• Signs join the terms: −2x + 6. The page shows the answer as MathML; Copy answer gives this text.

Assumptions

• Like terms are recognised by their written form only; no identities (such as sin² + cos² = 1) are used.

Worked examples by hand

3x + 7 + 4x + 5 (OpenStax Prealgebra 2e, section 2.2, Example 2.21). The x terms: 3x + 4x = 7x. The numbers: 7 + 5 = 12. Answer 7x + 12.

7x² + 8x − x² − 4x (Example 2.22). The x² terms: 7x² − x² = 6x². The x terms: 8x − 4x = 4x. Answer 6x² + 4x.