Factoring Polynomials

Introduction to Factoring

Factoring is a fundamental skill in algebra that involves rewriting a polynomial as a product of simpler expressions. This process is the reverse of multiplication and is essential for solving equations, especially quadratic equations. In this section, we focus on two primary factoring techniques: factoring out the Greatest Common Factor (GCF) and factoring by grouping.

Greatest Common Factor (GCF)

Definition and Concept

• Factor (noun): A number or expression that divides another number or expression exactly, without leaving a remainder. For example, 12 is a factor of 24 because .

• Factor (verb): To write a number or expression as a product of its factors. For example, factoring 24 gives .

• Factoring a Polynomial: Rewriting a sum or difference of terms as a product of expressions.

• Greatest Common Factor (GCF): The largest factor that divides each term of a set of numbers or expressions.

Finding the GCF of Numbers

• List the factors of each number.

• The GCF is the largest factor common to all numbers.

• It is efficient to start with the smallest number and check its factors.

Example: Find the GCF of 36, 42, and 12.

• Factors of 12: 1, 2, 3, 4, 6, 12

• Check which of these divides 36 and 42. 6 is the largest such number.

• GCF = 6

Finding the GCF with Variables

• For each variable, choose the smallest exponent present in all terms.

• The GCF cannot have a higher exponent than any term in the set.

Example: Find the GCF of and .

• Numerical GCF: 2 (since 2 divides both 18 and 32)

• Variable GCF: (smallest exponent)

• GCF = 2x^2

Factoring Out the GCF

• Rewrite the polynomial as the product of the GCF and another polynomial.

• Each term is divided by the GCF to find the remaining factor.

Example: Factor .

• GCF of coefficients: 9

• GCF of variables:

• GCF:

• Divide each term by :

• Factored form:

Example: Factor .

• GCF:

• Factored form:

Factoring with Multiple Variables

Example: Factor .

• GCF:

• Factored form:

Factoring Out a Negative GCF

• Factoring out a negative changes the signs of all terms inside the parentheses.

• This is useful when the leading term is negative.

Example: Factor .

• GCF:

• Factored form:

Factoring Binomials with a Common Binomial Factor

• Sometimes, the GCF is a binomial expression.

Example: Factor .

• GCF:

• Factored form:

Factoring by Grouping

Definition and Process

Factoring by grouping is a technique used when a polynomial has four or more terms and no single GCF for all terms. The terms are grouped into pairs, and the GCF is factored from each pair. If the resulting binomials are the same, they can be factored further.

• Group the terms into two pairs.

• Factor the GCF from each pair.

• If the resulting binomials are identical, factor out the common binomial.

Example 1: Factoring by Grouping

Factor .

• Group:

• GCF of first group:

• GCF of second group: $7$

• Factored:

• Common binomial:

• Final factored form:

Example 2: Factoring by Grouping

Factor .

• Group:

• GCF of first group:

• GCF of second group: $3$

• Factored:

• Common binomial:

• Final factored form:

Factoring by Grouping with Sign Changes

• If the binomials do not match, factor out a negative from one group to change the sign.

Example: Factor .

• Group:

• GCF of first group:

• GCF of second group: (to match signs)

• Factored:

• Common binomial:

• Final factored form:

Factoring by Grouping with Rearrangement

• If grouping does not immediately produce matching binomials, rearrange the terms and try again.

• Switching the order of terms can help create matching binomials.

Checking Factored Forms

• Always check your answer by multiplying the factors to ensure you obtain the original polynomial.

Summary Table: Factoring Techniques Covered

Additional info: Factoring is a foundational skill for solving quadratic equations and simplifying rational expressions, which are covered in later chapters.