Factoring Calculator

Factor expressions, polynomials, quadratics, and trinomials with clear, step-by-step solutions. Follow each factoring step from the original expression to the final answer.

Try an example to see it in action

The Factoring Calculator breaks an expression into its simpler factors and shows each step along the way, so you can check your work and see how the factorization builds. Type the problem, enter it with proper math notation, or handwrite and upload a photo. It works with numbers, algebraic expressions, and polynomials, and it can graph an expression when graphing adds useful context. For other types of math problems, you can use the AI Math Solver.

How to Use This Factor Calculator

1. Add Your Problem:

Type your expression in the Calculator tab using the basic on-screen keypad, switch to Math Input for proper exponents and symbols, or use Canvas to handwrite the expression or upload a photo of it. The same entry works whether your expression is a simple binomial or a longer polynomial.

2. Pick a Tool:

Open the Tools dropdown to choose your output. The default gives a step-by-step solution, which is what most factoring problems need. Pick Explain Like I'm 10 for a plainer walk-through when a particular step isn't clicking.

As a factoring calculator with steps, it shows each transformation — pulling out a common factor, then continuing — rather than just the final answer. When the entered expression is suitable for graphing, you can also select Create Graph to plot it.

3. Set Your Language:

Use the language dropdown labeled "Default" to set the output language. The steps and explanations appear in whatever language you choose.

4. Get Your Answer:

Click Solve to submit your expression. The selected result — step-by-step solution, simplified explanation, or graph — appears below.

How to Factor an Expression

Learning how to factor starts with reading the structure of the expression in front of you. A common factor, a recognizable pattern, or a group of terms might be your way in. There's no single method — here are the moves you'll use most often.

Factoring Out the GCF

When every term shares a common factor, pull it out first. For \(6x^2 + 9x\), both terms are divisible by \(3x\), so you factor out the GCF:

\[ 6x^2 + 9x = 3x(2x + 3) \]

This is GCF factoring, and it's the first move on most problems because it shrinks everything that follows. Knowing how to find the GCF — the largest factor shared by every term — saves work on nearly every expression you meet.

Factoring Trinomials

A trinomial like \(x^2 + 7x + 12\) factors into two binomials when you find a factor pair of \(12\) that adds to \(7\), giving \((x + 3)(x + 4)\). For \(ax^2 + bx + c\) with \(a \neq 1\), find a pair whose product is \(ac\) and whose sum is \(b\), then split the middle term and factor by grouping. If no such pair exists over the integers, the trinomial doesn't factor there — that's a property of the expression, not a mistake on your part. If you're working with a quadratic equation and need to continue to the solution, use the Quadratic Equation Calculator.

Factoring by Grouping

For four-term polynomials, group terms in pairs and factor a common factor out of each. With \(x^3 + 2x^2 + 3x + 6\):

\[ (x^3 + 2x^2) + (3x + 6) = x^2(x + 2) + 3(x + 2) = (x + 2)(x^2 + 3) \]

Factoring by grouping works when the two pairs share a common binomial factor.

Difference of Squares

Two perfect squares separated by a minus follow \(a^2 - b^2 = (a - b)(a + b)\). So \(9x^2 - 25\) becomes \((3x - 5)(3x + 5)\). The key is checking that both terms are perfect squares and that the operation between them is subtraction.

Perfect-Square Trinomials

A trinomial that's the square of a binomial follows \(a^2 + 2ab + b^2 = (a + b)^2\) or \(a^2 - 2ab + b^2 = (a - b)^2\). For example, \(x^2 + 6x + 9 = (x + 3)^2\). To recognize one, confirm the first and last terms are perfect squares and the middle term is twice the product of their roots.

Sum and Difference of Cubes

Cubes follow two patterns:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

So \(x^3 - 8\) factors as \((x - 2)(x^2 + 2x + 4)\). Notice that the quadratic factor here doesn't factor further over the reals.

Factoring Completely

After each step, ask whether any remaining factor can be broken down further over the number system you're working in. To factor completely means continuing until every factor is prime — nothing left splits into smaller meaningful factors. Some expressions, like \(x^2 + 1\) over the real numbers, simply have no further factorization, and that's a valid final answer.

What Factoring Means

Factoring rewrites an expression as a product of simpler factors. The same idea applies to numbers, algebraic expressions, and polynomials — the positive factors of \(24\) are \(1, 2, 3, 4, 6, 8, 12,\) and \(24\), while the factors of \(6x + 9\) are \(3\) and \((2x + 3)\). What's available always depends on the expression and the number system you're working in.

Two forms matter here. Expanded form sums the products, like \(x^2 + 5x + 6\); factored form writes them as a product, like \((x + 2)(x + 3)\). The two represent the same expression, just arranged differently.

Factoring an expression is also different from solving an equation. Factoring changes the form of an expression; solving an equation finds the values that make it true. For \(x^2 + 5x + 6 = 0\), factoring gives \((x + 2)(x + 3) = 0\), and the zero-product property yields \(x = -2\) or \(x = -3\). Once you've factored an equation, you can use the Solve For x Calculator to find the value of \(x\).

Worked Example: Factoring \(6x^3 + 15x^2 - 36x\)

Step 1: Factor out the GCF first.

\[ 6x^3 + 15x^2 - 36x = 3x(2x^2 + 5x - 12) \]

Every term shares a factor of \(3x\), so pull it out before anything else.

Step 2: Look at what remains.

\[ 2x^2 + 5x - 12 \]

The trinomial inside the parentheses can be factored further, so stopping here would leave the answer incomplete.

Step 3: Find two numbers that multiply to \(-24\) and add to \(5\).

\[ 8 \text{ and } -3, \quad \text{since } 8 \cdot (-3) = -24 \text{ and } 8 + (-3) = 5 \]

For \(ax^2 + bx + c\), you need a pair whose product is \(ac\) and whose sum is \(b\).

Step 4: Split the middle term and factor by grouping.

\[ 2x^2 + 8x - 3x - 12 = 2x(x + 4) - 3(x + 4) = (x + 4)(2x - 3) \]

Grouping reveals the shared binomial \((x + 4)\). The AC method reaches the same result; grouping simply makes the split visible.

Step 5: Combine everything.

\[ 6x^3 + 15x^2 - 36x = 3x(x + 4)(2x - 3) \]

The factors \(x\), \((x + 4)\), and \((2x - 3)\) are all linear, and the remaining \(3\) is just a numerical coefficient, so the expression is fully factored over the integers. Stopping back at Step 1 would have hidden the trinomial that still needed work.

Common Mistakes

Forgetting the GCF. Jumping straight to a trinomial while a common factor like \(3x\) is still there makes the rest harder and leaves the answer incomplete.

Choosing the wrong factor pair. For \(2x^2 + 5x - 12\), picking a pair that multiplies to \(12\) instead of \(ac = -24\) gives the wrong middle term. The pair's product must match \(ac\), not \(c\).

Misapplying a pattern. Writing \(x^2 - 9\) as \((x - 3)^2\) confuses a difference of squares with a perfect square. A minus between squares factors as \((a - b)(a + b)\), not as a square.

Stopping before factoring completely. Leaving \(3x(2x^2 + 5x - 12)\) as the final answer misses the trinomial inside, which still factors. Always re-check each remaining factor.

Assuming every expression factors. Some expressions, like \(x^2 + x + 1\) or \(x^2 + 1\) over the reals, have no factorization into simpler polynomial factors. Not finding factors can be the correct conclusion.

Losing track of negative signs. When you pull out a negative number, you must remember to flip the signs of the remaining terms inside the parentheses.

When Can't an Expression Be Factored?

Not every expression has a useful factorization over the integers. Some polynomials are already irreducible in the form you're working with, while others may factor only if you allow fractions, irrational numbers, or complex numbers.

For example, \(x^2+4\) doesn't factor into real linear factors, while \(x^2-4\) does because it follows the difference-of-squares pattern. Knowing when to stop is part of factoring correctly.

Questions

Frequently Asked Questions About Factoring

What is factoring?

Factoring is rewriting an expression as a product of simpler factors. For example, \(x^2 + 5x + 6\) becomes \((x + 2)(x + 3)\). The factored form and the original expression represent the same quantity, just arranged differently.

How do I factor an expression?

Start by checking for a greatest common factor across all terms, then look for a recognizable pattern like a difference of squares or a perfect-square trinomial. The method depends on the expression's structure, so there's no single order that works every time. If no common factor or pattern fits, the expression may not factor over the numbers you're using.

How do I factor a polynomial?

Look at how many terms it has. A two-term polynomial might be a difference of squares or cubes; a trinomial often factors into two binomials; a four-term polynomial is usually handled by grouping. Pull out any common factor first, then apply the method that matches the structure.

How do I factor trinomials?

For a trinomial like \(x^2 + bx + c\), find a factor pair of \(c\) that adds to \(b\). For \(ax^2 + bx + c\) with \(a \neq 1\), find a pair whose product is \(ac\) and whose sum is \(b\), then split the middle term and factor by grouping. If no such pair exists over the integers, the trinomial doesn't factor there.

How do I factor by grouping?

Split a four-term polynomial into two pairs and factor a common factor out of each. If the pairs share a common binomial, pull it out. For \(x^3 + 2x^2 + 3x + 6\), this gives \((x + 2)(x^2 + 3)\).

How do I find the GCF when factoring?

List the factors of each term and take the largest one they share, including variables raised to the lowest power that appears in every term. For \(6x^3 + 15x^2 - 36x\), the numerical GCF is \(3\) and the variable part is \(x\), so the GCF is \(3x\). Factoring out the GCF first usually makes the rest of the problem easier.

What is the difference between factoring and solving?

Factoring changes the form of an expression, rewriting it as a product of factors. Solving an equation finds the values that make it true. Factoring is often a step within solving, but the two are not the same.

How do I factor a quadratic equation?

To factor a quadratic equation, first rewrite it so one side is zero, then factor the quadratic expression and apply the zero-product property. For \(x^2 + 5x + 6 = 0\), this gives \((x + 2)(x + 3) = 0\), so \(x = -2\) or \(x = -3\). This works only when the quadratic actually factors; otherwise you'd use the quadratic formula or completing the square.

When can an expression not be factored?

An expression has no further factorization when no common factor, recognizable pattern, or factor pair works over the number system you're using. For example, \(x^2 + 1\) doesn't factor over the real numbers, and \(x^2 + x + 1\) has no integer factorization. This is a property of the expression, not an error in your work.

How does a factor calculator with steps help?

Seeing the individual steps reveals the specific mathematical rules applied between the expanded form and the final answer. This prevents you from simply memorizing the final result without understanding the logic.

When should I use an equation factorization calculator instead?

You use an equation solver when your expression contains an equals sign and you need to find the value of a variable. A standard factoring expressions calculator only changes the format of the mathematical phrase without solving for \(x\).

What is the difference of squares pattern?

It's the identity \(a^2 - b^2 = (a - b)(a + b)\), which applies whenever you're subtracting one perfect square from another. It does not apply to a sum of squares, since \(a^2 + b^2\) doesn't factor this way over the integers.

What's the difference between factored form and expanded form?

Expanded form is an expression written out as a sum of terms, like \(x^2 + 5x + 6\). Factored form is the same expression written as a product, like \((x + 2)(x + 3)\). Both represent the same value for any given \(x\); they're just written differently.

How do I know when an expression is factored completely?

An expression is completely factored when none of its remaining factors can be broken down any further within the number system you're using. This usually means checking each factor for a common term, a recognizable pattern, or a further trinomial or grouping structure.