How to factor

What is Factoring?

Factoring, in mathematics, is the process of breaking down a number or an algebraic expression into a product of smaller parts, called factors. Think of it as the opposite of multiplication. If multiplication combines numbers to make a larger number, factoring separates a number into the numbers that were multiplied to create it.

Understanding Factors of Numbers

When we talk about factoring integers (whole numbers), we are looking for two or more whole numbers that, when multiplied together, give us the original number. These numbers are called the factors of the original number.

Example: Factoring the number 12

Let's take the number 12. To find its factors, we ask ourselves: "What pairs of whole numbers multiply to give 12?"

• 1 x 12 = 12. So, 1 and 12 are factors.
• 2 x 6 = 12. So, 2 and 6 are factors.
• 3 x 4 = 12. So, 3 and 4 are factors.

Therefore, the factors of 12 are 1, 2, 3, 4, 6, and 12. Note that we list each factor only once.

Prime vs. Composite Numbers

• Prime Numbers: A prime number is a whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, 13, etc. The number 2 is the only even prime number.
• Composite Numbers: A composite number is a whole number greater than 1 that has more than two factors. Examples include 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6), 9 (factors: 1, 3, 9), and 12 (factors: 1, 2, 3, 4, 6, 12).

Factoring Algebraic Expressions

Factoring extends beyond simple numbers into the realm of algebra. In algebra, factoring involves rewriting a polynomial (an expression with variables and coefficients) as a product of simpler polynomials or monomials (expressions with a single term).

Common Factoring Techniques

There are several methods for factoring algebraic expressions:

1. Greatest Common Factor (GCF)

This is often the first step in factoring any polynomial. The GCF is the largest factor that two or more terms share. To factor out the GCF, you identify the GCF of the terms and then divide each term by the GCF. The GCF is then placed outside parentheses, with the results of the division placed inside.

Example: Factor 4x + 8

• The GCF of 4x and 8 is 4.
• Divide 4x by 4: x
• Divide 8 by 4: 2
• So, 4x + 8 can be factored as 4(x + 2).

2. Factoring Trinomials (Quadratic Expressions)

A trinomial is a polynomial with three terms, often in the form ax² + bx + c. Factoring trinomials can be more complex and involves finding two binomials that multiply to give the original trinomial.

Example: Factor x² + 5x + 6

We need to find two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the x term). These numbers are 2 and 3.

• So, x² + 5x + 6 can be factored as (x + 2)(x + 3).

There are various methods for factoring trinomials, including trial and error, grouping, and using specific formulas for difference of squares or sum/difference of cubes.

3. Difference of Squares

This pattern applies to binomials where two perfect squares are subtracted. The formula is: a² - b² = (a - b)(a + b).

Example: Factor x² - 9

• Here, a² = x² (so a = x) and b² = 9 (so b = 3).
• Using the formula, x² - 9 = (x - 3)(x + 3).

Why is Factoring Important?

Factoring is a fundamental skill in mathematics with numerous applications:

• Simplifying Expressions: Factoring allows you to simplify complex algebraic expressions, making them easier to work with.
• Solving Equations: It's a key technique for solving polynomial equations, especially quadratic equations. If you can factor an equation like ax² + bx + c = 0 into (px + q)(rx + s) = 0, you can easily find the solutions by setting each factor equal to zero.
• Graphing Functions: Understanding the factored form of a polynomial can help in finding its roots (where the graph crosses the x-axis), which is essential for graphing.
• Working with Fractions: Factoring is used to simplify rational expressions (algebraic fractions).

Mastering factoring techniques provides a powerful toolset for tackling more advanced mathematical concepts.