Factoring

Introduction to Factoring

Factoring is a fundamental process in algebra that involves writing a polynomial as the product of two or more simpler polynomials. This process is the reverse of multiplication and is essential for simplifying expressions and solving equations.

• Factoring reverses the process of multiplying polynomials.

• Factoring helps in solving polynomial equations and simplifying algebraic expressions.

Definition of Key Terms

• Polynomial: An algebraic expression consisting of terms with variables raised to whole number exponents.

• Factor: A number or expression that divides another number or expression evenly.

• Greatest Common Factor (GCF): The largest term that is a factor of all terms in a polynomial.

Factoring Out the Greatest Common Factor (GCF)

Understanding the GCF

The first step in factoring a polynomial is to identify and factor out the greatest common factor (GCF) from all terms. The GCF is the largest expression that divides each term of the polynomial without a remainder.

• To find the GCF, compare the coefficients and variable parts of each term.

• Factoring out the GCF simplifies the polynomial and prepares it for further factoring.

Procedure for Factoring Out the GCF

1. Identify the GCF of all terms in the polynomial.

2. Rewrite the polynomial as the product of the GCF and the remaining terms.

3. Check your work by multiplying the factors to ensure you obtain the original polynomial.

Examples

• Example 1: Factor out the GCF from .

  • The GCF of 15 and 20 is 5, and the GCF of and is $x$.

  • Factored form:

• Example 2: Factor out the GCF from .

  • The GCF of 25 and 30 is 5, and the GCF of and is $y^2$.

  • Factored form:

• Example 3: If there is no common factor other than 1, the polynomial cannot be factored further using GCF.

Factoring Out a Binomial Factor

Sometimes, the GCF is a binomial (an expression with two terms). In such cases, factor out the binomial from each term.

• Example:

Factoring Out a Negative Common Factor

It is possible to factor out a negative common factor, which can help simplify the expression or make further factoring easier.

• Example: or

Factoring by Grouping

Introduction to Factoring by Grouping

Factoring by grouping is a method used when a polynomial has more than three terms and the GCF of all terms is 1. This technique involves grouping terms in pairs (or other logical groupings) so that each group has a common factor.

• Factoring by grouping is especially useful for polynomials with four or more terms.

• Each group should have a common factor that can be factored out.

Procedure for Factoring by Grouping

1. Step 1: Group terms. Collect the terms into groups so that each group has a common factor.

2. Step 2: Factor within the groups. Factor out the common factor in each group.

3. Step 3: Factor the entire polynomial. If each group now has a common factor, factor it out. If not, try a different grouping.

4. Always check the factored form by multiplying.

Examples

• Example 1: Factor by grouping.

  • Group:

  • Factor within groups:

  • Factor the entire polynomial:

• Example 2: Factor by grouping.

  • Group:

  • Factor within groups:

  • Factor the entire polynomial:

Factoring Out a GCF before Factoring by Grouping

Sometimes, it is necessary to factor out the GCF first before applying the grouping method.

• Example: Factor .

  • First, factor out the GCF (7):

  • Then, group:

  • Factor within groups:

  • Factor the entire polynomial:

Summary Table: Factoring Methods