Factor Calculator

One box, one click. Type the polynomial you want to factor — the factor calculator form is pre-filled with $x^2 + 5x + 6$. Press Calculate, and the factor calculator returns the complete factored form instantly. No menus, no setup — the factor calculator reads standard typing with ^ for powers, so x^2 + 5x + 6, x^2 - 9, and 2x^2 + 7x + 3 all work exactly as written. The example chips below the box swap in ready-made expressions, and every answer from the factor calculator comes as typeset math you can copy.

The factor calculator runs entirely in your browser — the factoring happens the moment you click, with no waiting and no request leaving the page. Type any of these into the factor calculator and watch the pattern land:

• A trinomial — $x^2 + 5x + 6$. The factor calculator finds the two binomials whose product rebuilds it.
• A difference of squares — $x^2 - 9$. The factor calculator spots $a^2 - b^2$ and splits it into $(a-b)(a+b)$.
• Something with a common factor — $12x^3 + 18x^2$. The factor calculator pulls out the greatest common factor first.
• A fourth-degree expression — $x^4 - 16$. The factor calculator keeps going until nothing left will factor, which is what factoring completely means.

This page is more than a static factor calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. "Factor $2x^2 + 7x + 3$ step by step." The AI shows every move — the same moves the factor calculator makes in one click.
• Take a photo of your homework. The AI reads the expression and factors it, explaining the pattern it matches.
• Ask about the ideas, not just the answer. "What is the greatest common factor?" or "How do I know when I'm done factoring?" — the AI answers beside the factor calculator form.
• Check your hand work. Factor by hand, then run the same expression through the factor calculator and compare. The two answers must match exactly.

Factoring reverses multiplying. To factor a number or an expression means to rewrite it as a product of simpler pieces — its factors — that multiply back to the original. Every factoring claim is checked by one move: multiply the factors together and confirm you rebuild what you started with. The factor calculator makes that check its whole job.

For numbers, a factor is a value that divides exactly, with no remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12, because $12 = 1 \times 12 = 2 \times 6 = 3 \times 4$. For expressions the idea is the same with letters: $x^2 - 9$ factors as $(x-3)(x+3)$, because multiplying those two binomials rebuilds $x^2 - 9$ exactly. FOIL the check yourself — $x^2 + 3x - 3x - 9$ — the middle terms cancel and the original returns. A factor calculator is a machine for finding those pieces.

$x^2 - 9 = (x-3)(x+3)$

Why factor at all? Because products expose solutions. The equation $x^2 - 9 = 0$ looks stuck until it factors; written as $(x-3)(x+3) = 0$ it reads off $x = 3$ and $x = -3$ immediately — when a product equals zero, at least one factor must be zero. That zero-product property is the reason the quadratic formula works and the reason every algebra course drills factoring. Fractions simplify through factors too: $\frac{x^2-9}{x+3}$ collapses to $x-3$ the moment you see the shared factor. The factor calculator is the fastest route to seeing those hidden pieces.

Prime numbers are the end of the factoring line for whole numbers. A prime — 2, 3, 5, 7, 11 — has exactly two factors: 1 and itself. Every composite number factors all the way down into primes, in exactly one way, which is why primes are called the building blocks of the integers. The next sections walk the full method the factor calculator runs: common factors first, then patterns, then trinomials — and the factor tree picture shows the number side of the same idea.

The factor calculator follows a fixed checklist on every expression, and the checklist is short enough to memorize. Run it in order — each step either makes progress or rules out a method:

1. Pull out the greatest common factor (GCF) first. Every term must share it. For $12x^3 + 18x^2$ the GCF is $6x^2$, leaving $6x^2(2x + 3)$. Nothing else works well until this is done.
2. Count the terms. Two terms: look for a difference of squares ($a^2 - b^2$) or a sum/difference of cubes. Three terms: factor the trinomial — the pattern section below shows both shapes. Four terms: try grouping — pair the terms and factor each pair.
3. Match the pattern. Perfect squares of the form $a^2 - b^2$ split as $(a-b)(a+b)$; trinomials split into two binomials whose product and sum hit the two coefficients.
4. Check each factor again. This is the step everyone skips — and the step that separates an answer from the complete answer. $x^4 - 16$ factors to $(x^2-4)(x^2+4)$, but the $x^2 - 4$ factors again. Keep going until every piece is prime over the integers.
5. Verify by multiplying back. Expand your answer. If it rebuilds the original expression, the factoring is correct — the factor calculator runs exactly this check before showing a result.

That checklist is the whole subject. Each step has its own section on this page with worked examples: the GCF step, the difference of squares, trinomials, and the meaning of factor completely. Run any step's example through the factor calculator form above — the factor calculator confirms each intermediate factoring, so you can check your work at every stage, not just at the end.

The greatest common factor (GCF) of two terms is the largest factor they share — the biggest piece that divides every term exactly. Finding it is always step one of factoring, because expressions with a common factor hide their remaining structure until it is pulled out. The factor calculator runs this move before anything else on every input.

For plain numbers, list the prime factorizations and keep what overlaps. Take 48 and 180:

• $48 = 2^4 \cdot 3$
• $180 = 2^2 \cdot 3^2 \cdot 5$
• The overlap is $2^2 \cdot 3 = 12$ — so GCF$(48, 180) = 12$, and indeed $48 = 12 \cdot 4$ and $180 = 12 \cdot 15$.

For expressions, the GCF takes the greatest common numerical factor times the lowest power of each shared variable. In $12x^3 + 18x^2$: the numbers give GCF$(12, 18) = 6$, the shared variable part is $x^2$ (the lower power), so:

$12x^3 + 18x^2 = 6x^2(2x + 3)$

Divide each term by the GCF to fill the parentheses — $12x^3 \div 6x^2 = 2x$ and $18x^2 \div 6x^2 = 3$ — then multiply back to check. The check is not optional: $6x^2 \cdot 2x = 12x^3$ and $6x^2 \cdot 3 = 18x^2$, the original terms exactly.

The GCF answers a classic word problem too: split a 48-meter rope and a 180-meter rope into equal pieces of the greatest possible length — each piece is 12 meters, the GCF. The same question with $12x^3 + 18x^2$ has the same answer: the GCF is the largest single thing you can hand out to every term. Type $12x^3 + 18x^2$ into the factor calculator above and the GCF step is the first thing you'll see in the result.

For a whole number, complete factoring means prime factorization: split the number until every piece is prime. The classic picture is the factor tree — split each composite into two factors, circle the primes, and read the leaves. The tree below factors 72:

• $72 = 8 \cdot 9$
• $8 = 2 \cdot 2 \cdot 2$ and $9 = 3 \cdot 3$
• Every leaf is prime, so $72 = 2^3 \cdot 3^2$ — the complete prime factorization.

The tree can branch differently — $72 = 6 \cdot 12$ works just as well — but the leaves always land on the same primes with the same counts. That uniqueness is the Fundamental Theorem of Arithmetic: one number, one prime factorization. The factor calculator's number-side answers rest on it.

The prime factorization generates everything: all factors of 72 come from choosing powers from each prime — $2^a 3^b$ with $a \in \{0,1,2,3\}$ and $b \in \{0,1,2\}$ — giving $(3+1)(2+1) = 12$ factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The GCF of two numbers is the shared part of their prime factorizations, as the 48-and-180 example showed. And a number is a perfect square exactly when every prime exponent is even — $72 = 2^3 \cdot 3^2$ is not, but $144 = 2^4 \cdot 3^2$ is.

Ask the AI box for the prime factorization of any number — "factor 360 into primes" — and it draws the tree step by step. For polynomials, the same game plays with letters, and the next two sections cover the two patterns the factor calculator uses most: differences of squares and trinomials.

The fastest pattern in factoring: a difference of squares is $a^2 - b^2 = (a-b)(a+b)$ — two perfect squares with a minus sign between them. It works because the middle terms of the product cancel: $(a-b)(a+b) = a^2 + ab - ab - b^2$. Spotting it is pure pattern recognition:

• $x^2 - 9 = (x-3)(x+3)$ — here $a = x$, $b = 3$.
• $4x^2 - 25 = (2x-5)(2x+5)$ — $a = 2x$, $b = 5$; squares hide behind coefficients too.
• $x^2 + 9$ — the same two squares with a plus sign — is not a difference of squares and does not factor over the real numbers. The minus sign is essential.

Now the step that separates a finished answer from a half-finished one. Factoring completely means factoring until no factor can be factored again. Watch $x^4 - 16$, which wears two disguises:

1. First pass — it is a difference of squares with $a = x^2$, $b = 4$: $x^4 - 16 = (x^2 - 4)(x^2 + 4)$.
2. Check each factor — $x^2 + 4$ is done (a sum of squares does not factor), but $x^2 - 4$ is another difference of squares: $x^2 - 4 = (x-2)(x+2)$.
3. Complete answer: $x^4 - 16 = (x-2)(x+2)(x^2+4)$ — three factors, none of which factors again.

Stopping after step 1 is the most common factoring error on tests, and the factor calculator never makes it — the re-check step in its checklist loops until every factor is prime. Type $x^4 - 16$ into the factor calculator above and compare: you get the complete three-factor form, not the intermediate one. A practical habit: after any factoring, scan each parenthesis and ask "does this factor again?" — when every answer is no, you have factored completely.

A trinomial $ax^2 + bx + c$ factors into two binomials — when it factors at all. The simple case, $a = 1$, is a product-and-sum hunt: find two numbers that multiply to $c$ and add to $b$. For $x^2 + 5x + 6$ — the expression sitting in the factor calculator form on this page — the hunt is for a pair with product 6 and sum 5:

• 1 and 6: product 6, sum 7 — no.
• 2 and 3: product 6, sum 5 — yes.

So $x^2 + 5x + 6 = (x+2)(x+3)$. Multiply back to check: $x^2 + 3x + 2x + 6 = x^2 + 5x + 6$ — exact. Signs matter in the hunt: for $x^2 - 5x + 6$ the pair is $-2$ and $-3$ (product $+6$, sum $-5$), giving $(x-2)(x-3)$; for $x^2 + 5x - 6$ it is $-1$ and $6$, giving $(x-1)(x+6)$.

When $a \neq 1$, use the AC method — it turns the hard case into the easy one. Factor $2x^2 + 7x + 3$:

1. Compute $ac = 2 \cdot 3 = 6$. Hunt for a pair with product 6 and sum $b = 7$: it is 1 and 6.
2. Split the middle term along that pair: $2x^2 + x + 6x + 3$.
3. Group and factor each pair: $x(2x+1) + 3(2x+1)$.
4. Pull out the shared binomial: $(2x+1)(x+3)$.

Check by FOIL: $2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$ — exact. The trinomial is the workhorse pattern of Algebra 1, and the factor calculator runs the product-and-sum hunt (or the AC method) on every three-term input. Feed it $x^2 + 2x + 1$ as well — a perfect-square trinomial that lands on $(x+1)^2$, the same answer written with an exponent, and one more form the factor calculator reports.

Every algebra grader sees these three slips weekly. Each one leaves a wrong answer that looks finished, and the factor calculator catches all three instantly:

1. Forgetting the GCF first. Factor $3x^2 - 27$ by jumping straight to the difference of squares and you get $3(x-3)(x+3)$ only if you remember the 3 — many answers drop it and report $(x-3)(x+3)$, which multiplies back to $x^2 - 9$, not $3x^2 - 27$. The GCF is always step one precisely because nothing may be dropped. Multiply back and the missing 3 surfaces immediately.
2. Stopping before the factoring is complete. $x^4 - 16 = (x^2-4)(x^2+4)$ is a true statement — and an incomplete answer, because $x^2 - 4$ factors again. Test rubrics read "factor completely" as a instruction, not a suggestion; the full answer is $(x-2)(x+2)(x^2+4)$. The re-check-every-factor habit from the pattern section is the cure.
3. Sign slips in the product-and-sum hunt. For $x^2 + 5x - 6$, the pair needs product $-6$ and sum $5$ — that's $-1$ and $6$, yet answers with $(x+1)(x-6)$ appear constantly (product $-6$, sum $-5$ — wrong sum). Signs carry half the information in a trinomial; always verify the sum and the product separately, then multiply back.

The defense against all three is the same thirty-second move: multiply your answer out and compare with the original. The factor calculator automates the comparison — factor by hand, type the expression in, and any mismatch tells you exactly which slip you made. The practice problems below each end with that check.