Factorization Formula

The factorization formula is used to rewrite a mathematical expression as a product of simpler factors. It is an important algebraic technique that makes expressions easier to simplify, evaluate, and solve. Factoring is essentially the reverse of expanding a product.

What Is the Factorization Formula?

Factorization means expressing a number or algebraic expression as a product of two or more factors. For example:

x² + 5x + 6 = (x + 2)(x + 3)

When the factors are multiplied together, they give the original expression. This makes factorization useful for simplifying algebraic expressions and solving equations.

Common Factorization Formulas

Several standard identities are commonly used in factorization.

Difference of Squares:

a² − b² = (a − b)(a + b)

For example:

x² − 25 = (x − 5)(x + 5)

Perfect Square Trinomial:

a² + 2ab + b² = (a + b)²

a² − 2ab + b² = (a − b)²

For example:

x² + 6x + 9 = (x + 3)²

Sum of Cubes:

a³ + b³ = (a + b)(a² − ab + b²)

Difference of Cubes:

a³ − b³ = (a − b)(a² + ab + b²)

These identities provide quick ways to factor expressions that match their patterns.

How to Factor an Expression

Start by checking whether all terms have a common factor. If they do, take that factor outside the parentheses first. Then look for a familiar pattern, such as a difference of squares or a perfect square trinomial.

For example, consider:

6x² + 9x

The greatest common factor is 3x, so:

6x² + 9x = 3x(2x + 3)

The expression is now completely factored.

Difference of Squares Formula

The difference of squares is one of the most useful factorization patterns. It applies when two perfect squares are being subtracted:

a² − b² = (a − b)(a + b)

For example:

9x² − 16 = (3x − 4)(3x + 4)

This works because 9x² is the square of 3x and 16 is the square of 4.

Factoring Quadratic Expressions

A quadratic expression often has the form:

ax² + bx + c

When the expression can be factored, it may be rewritten as the product of two simpler expressions. For example:

x² + 7x + 12 = (x + 3)(x + 4)

To verify the result, multiply the two factors:

(x + 3)(x + 4) = x² + 7x + 12

Different quadratic expressions may require different methods, including common-factor extraction, grouping, the sum-product method, or special product formulas.

Important Factorization Tips

Always check for a common factor before applying another factorization formula. After factoring an expression, check whether any of the resulting factors can be factored further.

It is also important not to confuse the difference of squares with a sum of squares. The expression a² − b² factors into two binomials, while a² + b² does not factor into real binomials in the same way.