Factor The Following Expression Calculator

This calculator helps you factor polynomial expressions by identifying common factors. Whether you're studying algebra or need to simplify expressions for calculus, this tool provides step-by-step guidance and instant results.

How to Use This Calculator

Enter your polynomial expression in the input field below. The calculator will analyze the expression and provide the factored form. You can also select the type of factoring you want to perform if you know it in advance.

Tip

For best results, enter expressions in standard form (e.g., 3x² + 6x + 3) without spaces between terms. The calculator handles both positive and negative coefficients.

Step-by-Step Process

Enter your polynomial expression in the input field.
Select the type of factoring (if known).
Click "Calculate" to see the factored form.
Review the step-by-step solution if needed.
Use the result in your work or further calculations.

How Factoring Works

Factoring is the process of breaking down a polynomial into a product of simpler polynomials. This is useful for solving equations, simplifying expressions, and understanding the structure of polynomials.

General Factoring Principle

If a polynomial can be written as a product of two or more polynomials, then it is factorable. The most common methods are:

Factoring out the greatest common factor (GCF)
Factoring by grouping
Using special formulas (difference of squares, perfect square trinomials)
Factoring quadratics (ax² + bx + c)

Key Concepts

GCF: The largest polynomial that divides all terms of the expression.
Difference of Squares: a² - b² = (a + b)(a - b)
Perfect Square Trinomial: a² ± 2ab + b² = (a ± b)²

Common Types of Factoring

Different types of polynomials require different factoring techniques. Here are the most common types:

1. Factoring Out the GCF

This is the simplest form of factoring. You identify the largest term that divides all terms of the polynomial and factor it out.

Example

Factor: 6x² + 9x

Solution: GCF is 3x, so 3x(x + 3)

2. Factoring by Grouping

This method involves grouping terms in pairs and factoring each pair separately.

Example

Factor: xy + xz + yz + x

Solution: Group xy + xz and yz + x, then factor as x(y + z) + 1(yz + x)

3. Special Formulas

Some polynomials fit special patterns that can be factored using known formulas.

Difference of Squares

a² - b² = (a + b)(a - b)

Example: 9x² - 16 = (3x + 4)(3x - 4)

Perfect Square Trinomial

a² ± 2ab + b² = (a ± b)²

Example: x² + 6x + 9 = (x + 3)²

Factoring Examples

Here are some examples of how to factor different types of expressions:

Example 1: Factoring Out the GCF

Factor: 8x³ + 12x²

Solution: GCF is 4x², so 4x²(2x + 3)

Example 2: Factoring by Grouping

Factor: 2x²y + 4xy + 6xy + 12y

Solution: Group 2x²y + 4xy and 6xy + 12y, then factor as 2xy(x + 2) + 6y(x + 2) = 2(x + 2)(xy + 3y)

Example 3: Difference of Squares

Factor: 25a² - 49b²

Solution: (5a + 7b)(5a - 7b)

Example 4: Perfect Square Trinomial

Factor: 16x² - 24x + 9

Solution: (4x - 3)²

FAQ

What is the difference between factoring and expanding?

Factoring is the process of breaking down a polynomial into a product of simpler polynomials, while expanding is the process of multiplying out the factors to get back to the original polynomial. Factoring is typically used to simplify expressions, while expanding is often used to solve equations.

Can all polynomials be factored?

No, not all polynomials can be factored. Some polynomials, especially those with irrational roots, cannot be factored using real numbers. However, they can be factored using complex numbers or other advanced techniques.

What if my polynomial doesn't factor nicely?

If your polynomial doesn't factor neatly, it might be irreducible over the real numbers. In such cases, you can still factor it using complex numbers or leave it in its original form. The calculator will indicate if the expression cannot be factored further.

How do I know if I've factored correctly?

To verify your factoring, you can expand the factored form and see if it matches the original expression. For example, if you factor x² + 5x + 6 as (x + 2)(x + 3), expanding gives x² + 5x + 6, which matches the original.

Can this calculator handle negative coefficients?

Yes, the calculator can handle negative coefficients. Simply enter the expression with negative signs, and the calculator will process it correctly. For example, x² - 4x + 4 will be factored as (x - 2)².