How To Factor

Factoring in Standard American English

Factoring is a mathematical operation that involves writing a given expression as a product of two or more simpler expressions. This process is commonly used to simplify complex expressions, solve algebraic equations, and perform various mathematical operations.

Basic Principles of Factoring

The fundamental principle of factoring is to identify and group common factors within the given expression. A common factor is a term or variable that is divisible by all the other terms in the expression.

Steps in Factoring:

1. Identify the Greatest Common Factor (GCF): Find the largest factor that divides every term in the expression evenly. The GCF can be a single variable, a numerical coefficient, or a combination of both.

2. Factor out the GCF: Divide each term in the expression by the GCF. This results in an expression with each term containing at least one of the GCF’s factors.

3. Group Similar Factors: Identify any groups of terms that have common factors. These factors can be numerical coefficients, variables, or a combination.

4. Factor out the Common Factors: Divide each group of terms by the common factor. This results in an expression where each group of terms is now a product of simpler factors.

Types of Factoring

There are several methods of factoring commonly used in algebra:

1. Factoring by Common Factors: Identify and factor out the GCF.

2. Factoring by Grouping: Group terms with common factors and factor out those common factors.

3. Factoring Trinomials: Factor quadratic expressions of the form ax² + bx + c.

4. Factoring Difference of Squares: Factor expressions of the form a² – b².

5. Factoring Sum and Difference of Cubes: Factor expressions of the forms a³ + b³ and a³ – b³.

Examples of Factoring

Example 1: Factoring by Common Factors

Factor the expression: 12x³y² – 18xy

Solution:

1. GCF: 6xy
2. Divide each term by the GCF: 6x²y – 3
3. Factor out the GCF: 6xy(2x – 1)

Example 2: Factoring by Grouping

Factor the expression: x³ – 3x² – 4x + 12

Solution:

1. Identify groups: (x³ – 3x²) and (-4x + 12)
2. GCF of first group: x²
3. Divide and factor: x²(x – 3)
4. GCF of second group: 4
5. Divide and factor: 4(1 – x)
6. Final factored expression: (x² – 3)(1 – x)

Applications of Factoring

Factoring has various applications in mathematics and algebra, including:

1. Solving Algebraic Equations: Factoring can simplify equations and isolate variables for solving.
2. Simplifying Expressions: Factoring can reduce complex expressions into simpler forms, making them easier to analyze and manipulate.
3. Finding Roots and Intercepts: Factoring can be used to determine the roots (zeros) and intercepts of functions.
4. Geometry: Factoring can be helpful in solving geometry problems involving area, volume, and perimeter.

FAQ

1. What is the difference between factoring and expanding?

Factoring involves writing an expression as a product of simpler factors, while expanding involves multiplying out factors to obtain a larger expression.

2. How do I know if an expression is factorable?

Not all expressions are factorable. However, if an expression contains common factors or has a specific form (e.g., a trinomial), it may be possible to factor it.

3. Do I always need to factor out the GCF first?

Yes, factoring out the GCF is usually the first step to simplify factoring. This step reduces the overall size of the expression and makes it easier to identify other common factors.

4. What if I cannot factor an expression any further?

If no common factors can be identified after using factoring techniques, the expression is considered prime and cannot be factored further.

5. Are there other methods of factoring besides the ones mentioned?

Yes, there are other advanced factoring techniques, such as factoring by substitution or using the quadratic formula. These methods are typically used for more complex expressions.