Factoring Calculator
Factor a whole number to see its factor pairs, full factor list, and prime-factor tree(plus the GCF and LCM of up to three numbers), or switch modes to factor a trinomialax² + bx + c using the AC method, difference of squares, or perfect-square shortcuts — with every step shown. Covers the factoring unit taught in grade 9/10 math across Canadian provinces: 48 factors as 2⁴ × 3; x² − 5x + 6 factors as (x − 2)(x − 3).
2^4 × 3
Prime factorization of 48
All factors (10): 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factor pairs: 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8
Show the prime-factor tree, step by step
GCF of 48, 60: 12
Show GCF steps
LCM of 48, 60: 240
Show LCM steps
Trinomial factoring works over the integers only — some trinomials have irrational or complex roots and cannot be factored this way. How we calculate →
Factoring a number: divisor pairs and the prime-factor tree
To factor a whole number, list every pair of numbers that multiply to it. 48 has factor pairs 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8 — 10 factors in total: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Breaking a number down into prime factors (the smallest building blocks, only divisible by 1 and themselves) uses a factor tree: repeatedly divide by the smallest prime that fits, until only primes are left. For 48: 48 → 2 × 24 → 2 × 2 × 12 → 2 × 2 × 2 × 6 → 2 × 2 × 2 × 2 × 3, so the prime factorization is 2^4 × 3.
GCF and LCM: two numbers, two different questions
The greatest common factor (GCF) is the largest number that divides two (or more) numbers evenly — it's what's left after cancelling every shared prime factor. GCF(48, 60) = 12, found with the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is 0.
The least common multiple (LCM) is the smallest number both are factors of — useful for adding fractions with different denominators. LCM and GCF are linked by a simple identity: LCM(a, b) = (a × b) ÷ GCF(a, b), so once you have the GCF, the LCM is one division away.
Factoring a trinomial: pull out the GCF first, always
Before trying anything else, check whether all three terms of 1x² − 5x + 6 share a common factor greater than 1 — skipping this step is the single most common factoring mistake, because it leaves an "unfactorable-looking" trinomial that's actually just missing its GCF.
Two special patterns are worth spotting immediately: a difference of squares (a² − b² = (a−b)(a+b), only when the middle term is missing) and a perfect square trinomial (when the discriminant b²−4ac is exactly 0 and the leading coefficient is a perfect square) — both factor faster than the general method below.
The AC method: factoring the general trinomial
For a trinomial that isn't a GCF, difference-of-squares, or perfect-square case, the AC method finds two numbers that multiply to a × c and add to b, then uses them to split the middle term so the four-term expression can be grouped and factored in pairs. For 6x² + 5x − 6: a × c = -36, and the pair -4 and 9 multiplies to -36 and adds to 5 — giving (3x − 2)(2x + 3).
If no integer pair multiplies to a×c and adds to b, the trinomial's discriminant (b² − 4ac) isn't a perfect square, which means the roots are irrational (or complex) — that trinomial genuinely cannot be factored over the integers, and this calculator says so instead of forcing a wrong answer.
Frequently asked questions
What are the factors of a number?
The factors of a number are every whole number that divides it evenly (no remainder). For example, the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 — each one pairs with another factor (e.g. 6 × 8 = 48) to multiply back to the original number.
How do you find the prime factorization of a number?
Divide repeatedly by the smallest prime number that fits, replacing the number with the quotient each time, until only 1 is left. For 48, that's 2 × 2 × 2 × 2 × 3, written as 2^4 × 3.
What is the GCF (greatest common factor)?
The GCF of two or more numbers is the largest number that divides all of them evenly. GCF(48, 60) = 12, found by repeatedly dividing the larger number by the smaller and keeping the remainder (the Euclidean algorithm) until the remainder hits 0.
What is the difference between GCF and LCM?
GCF is the largest number that divides evenly INTO both numbers; LCM is the smallest number that both numbers divide evenly INTO. They're related: LCM(a,b) = (a × b) ÷ GCF(a,b).
How do you factor a trinomial like ax² + bx + c?
First pull out any common factor from all three terms. Then check for a difference of squares or perfect square trinomial pattern. If neither applies, use the AC method: find two numbers multiplying to a×c and adding to b, split the middle term using them, then factor by grouping.
What if a trinomial can't be factored?
Not every trinomial factors over the integers. Check the discriminant (b² − 4ac): if it isn't a perfect square (or is negative), the roots are irrational or complex, and no combination of integer binomials will multiply back to the original trinomial — the quadratic formula is then the only way to find the exact roots.
Is factoring by grouping the same as the AC method?
They're two names for the same overall process. The AC method is the step of finding the two numbers (that multiply to a×c and add to b); factoring by grouping is what you do with them next — splitting the middle term into two terms and pulling a common binomial out of each half.
Researched & verified by the Calcuris Data & Research Team. How we build and check our tools →