LESS GUESSING. MORE UNDERSTANDING.

Factoring Calculator

Find the factors. Follow the steps. Understand the why.

A little help with algebra.
Every step, always free.

Worked example

FACTORED FORMQuadratic trinomial

(x + 1)(x + 4)

from x² + 5x + 4

Let’s break it down

THE HOW + THE WHY

1
Find what the two numbers need to do
a + b = 5 and a · b = 4
Expanding (x + a)(x + b) gives x² + (a + b)x + ab. The two numbers must add to 5 and multiply to 4.
2
Choose the matching factor pair
1 + 4 = 5 and 1 · 4 = 4
The positive factor pairs of 4 are (1, 4) and (2, 2). Only 1 and 4 also add to 5.
3
Write the two factors
x² + 5x + 4 = (x + 1)(x + 4)
Substitute 1 and 4 for a and b. These linear factors cannot be factored further.

Check by expanding

Verified

(x + 1)(x + 4) = x² + 4x + x + 4 = x² + 5x + 4

We’re back to the original expression. It checks out.

MAKE THE PATTERNS CLICK

What is factoring?

Factoring rewrites a number or an expression as a product of simpler pieces, called factors. For example, 12 = 3 × 4 and x² − 9 = (x − 3)(x + 3). Multiply the factors back together and you get exactly where you started. Factoring changes the form of the expression, not its value.

A polynomial combines terms using addition or subtraction. A term might be a constant, a variable, or a coefficient multiplied by variables with whole number exponents. In 3x² + 6x, the coefficients are three and six. Looking at each term separately helps you notice the shared factor before choosing a more specific pattern.

Terms are added or subtracted; factors are multiplied. Parentheses let a whole expression act as one factor. In the square difference above, the answer has two factors, each containing two terms.

This factoring calculator explains each algebraic operation. Try predicting the next step before reading on. Enter a positive integer such as 48 to see its factors, factor pairs, and prime factorization. For a polynomial, compare your work with the solution and its expansion check.

How to Factor a Polynomial Step by Step

Start by combining like terms: terms with exactly the same variables and exponents. Put the polynomial in a familiar order, usually from highest to lowest power. Keep a negative sign attached to its term. This makes the number of terms and the leading coefficient easier to see, without changing the expression you need to factor.

Next, check for the greatest common factor. After taking it out, count the terms left inside the parentheses. Two terms may suggest a difference of squares or a cube pattern; three may suggest a quadratic or perfect square. These are clues, not guarantees. Check the exponents, coefficients, and signs against the pattern before using it.

For a quadratic with leading coefficient one, look for two numbers whose product is the constant and whose sum is the middle coefficient. With another leading coefficient, splitting the middle term can turn the quadratic into four terms that factor by grouping. If no integer pair works, do not invent one: the expression may need another method or may be irreducible over rational coefficients.

A positive product means both numbers have the same sign; their sum determines which sign. A negative product means opposite signs: the number with the greater absolute value determines the sum’s sign. Equal absolute values give a sum of zero. Check both product and sum before choosing the factors.

With multiple variables, compare each letter’s powers separately to find a common factor. An entire variable expression can stand for one quantity in a pattern. For a difference of squares, identify the two quantities being squared, then write their sum and difference.

Reading Your Factoring Solution

Factors can appear in any order without changing their product, and repeated factors can be written as powers. Use the factoring calculator’s expansion check to trace how every original term returns, including its sign and coefficient.

Finally, inspect every factor again and expand your answer. A factoring calculator can verify that a product matches the input, but equality alone does not prove that factoring is complete. A result marked partial still needs further work. Use the six patterns below to recognize what each step is trying to accomplish.

Six Common Factoring Patterns

Select an example to see the steps

Pattern	When and why it works	Try an example
01 Greatest common factorHow to find the GCF →	Use this when every term shares a number, a variable factor, or both. Divide each term by that shared factor, then place the factor outside parentheses: this reverses the distributive property. In the example, both terms share six times the variable.
02 Quadratic trinomialFactoring by grouping →	ax² + bx + c Use a sum and product pair for a quadratic that factors over integers. Expanding two linear factors explains why both conditions must match. Here, one and four multiply to four and add to five.
03 Difference of squaresRecognize the squares →	a² − b² = (a − b)(a + b) Use this for two squared quantities separated by subtraction. The opposite signs cancel the middle terms when you expand. In the example, three times the variable and five are the two quantities being squared.
04 Perfect square trinomial	a² ± 2ab + b² = (a ± b)² Use this when the outside terms are squares and the middle term is twice their square roots multiplied together, with either sign. That middle term comes from the two equal cross products. Here, three is the constant in the repeated factor.
05 Sum of cubes	a³ + b³ = (a + b)(a² − ab + b²) Use this when two cubes are added. The negative middle term in the quadratic factor cancels the extra terms produced during expansion. In the example, eight is the cube of two.
06 Difference of cubes	a³ − b³ = (a − b)(a² + ab + b²) Use this when one cube is subtracted from another. The linear factor uses subtraction, while every sign in the quadratic factor is positive before substituting values. Here, twenty seven is the cube of three.

Finding the Greatest Common Factor

For integer coefficients, find the greatest number dividing every coefficient, then use the smallest exponent shared by all terms for each variable. If a variable is absent from one term, it cannot belong to the common factor. For 6x² + 12x, taking out 6x leaves x + 2. Multiply back to check both terms.

Factoring a Difference of Squares

The squared quantities can contain more than one variable. In x² − y², use the sum and difference of the two variables as factors. A plus sign between the original squares does not fit this rule.

Factoring by Grouping

Split or rearrange terms so that two groups contain the same expression after their common factors are removed. For 2x² + 7x + 3, split the middle term into 6x + x; the groups then share x + 3.

CATCH THE SMALL SLIPS

Common Factoring Mistakes

Before copying a solution from the factoring calculator, compare it with your own steps. A wrong sign or a missing term often explains why an answer looks close but does not expand correctly.

Stopping after the first common factor. Taking out a shared number is useful, but the remaining polynomial may still factor. Recheck the parentheses for another common factor or a square pattern before calling the expression completely factored.
Matching the product but ignoring the sum. For a quadratic, the two numbers must satisfy both conditions. Write down their sum as well as their product; otherwise the middle coefficient will be wrong when you expand.
Losing a minus sign while grouping. Dividing a group by a negative factor changes every sign inside it. Distribute that negative factor back immediately, and check each term against the original group.
Using a difference rule on a sum. A sum of squares does not follow the difference of squares formula. Name the pattern before applying it, and check that the operation between the two squared quantities is subtraction.
Checking only the first and last terms. Multiply every term in one factor by every term in the other, then combine like terms. The middle terms are often where an incorrect sign or coefficient becomes visible.

After finding a mistake, enter the corrected expression in the factoring calculator and compare one step at a time. The goal is to understand why the factors work, so you can recognize the same structure in your next problem.

A FEW GOOD QUESTIONS

Let’s clear things up.

The small details that make
algebra make sense.

What is the difference between factoring and solving?

Factoring rewrites an expression as a product, such as x² + 5x + 4 = (x + 1)(x + 4). Solving finds values that make an equation true. If x² + 5x + 4 = 0, the factored form shows that x = −1 or x = −4.

What does it mean to factor completely?

Factor completely means taking out the greatest common factor and continuing until no remaining factor can be broken down over the chosen number system. This calculator looks for factors with rational coefficients. For example, x⁴ − 16 = (x − 2)(x + 2)(x² + 4); x² + 4 does not factor further over the rationals.

How do I factor when the leading coefficient is not 1?

For ax² + bx + c, find two integers whose product is ac and whose sum is b, when such a pair exists. Split the middle term using that pair, then factor by grouping. For 2x² + 7x + 3, use 6 and 1:

2x² + 6x + x + 3
= 2x(x + 3) + 1(x + 3)
= (2x + 1)(x + 3)