Factoring Polynomials

Introduction to Factoring

Factoring is the process of expressing a polynomial as a product of its factors. This is the reverse operation of expanding or distributing, which is covered in earlier sections. Factoring is a fundamental skill in algebra, especially for solving equations and simplifying expressions.

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

• Factor: A quantity that is multiplied by another to produce a given number or expression.

• Prime Polynomial: A polynomial that cannot be factored further over the integers.

Factoring Trinomials

Factoring Trinomials of the Form

To factor a trinomial, look for two binomials whose product gives the original expression. The process involves finding two numbers that multiply to (the product of the leading coefficient and the constant term) and add to (the coefficient of the middle term).

• Step 1: Check for a Greatest Common Factor (GCF) and factor it out if present.

• Step 2: Identify , , and in .

• Step 3: Find two numbers and such that and .

• Step 4: Rewrite the middle term as and factor by grouping.

Example 1: Factor

• Check: No GCF. , , .

• Find two numbers that multiply to and add to . The numbers are and .

• Rewrite:

• Group:

• Factor:

Example 2: Factor

• Check: No GCF. , , .

• Find two numbers that multiply to and add to $5 and .

• Rewrite:

• Group:

• Factor:

Factoring the Difference of Two Squares

Recognizing and Factoring

The difference of two squares is a special pattern that factors as follows:

• Both terms must be perfect squares and separated by a minus sign.

Example: Factor

• and $9$ are perfect squares.

• Factor:

Factoring Perfect Square Trinomials

Recognizing and Factoring

A perfect square trinomial is the result of squaring a binomial:

• Check if the first and last terms are perfect squares and the middle term is twice the product of their square roots.

Example: Factor

• and $9

• Factor:

Factoring by Grouping

Factoring Polynomials with Four Terms

When a polynomial has four terms, try factoring by grouping:

• Group the terms into two pairs.

• Factor out the GCF from each pair.

• If the remaining binomial factors are the same, factor them out.

Example: Factor

• Group:

• Factor:

Prime and Non-Factorable Polynomials

Recognizing Prime Polynomials

Some polynomials cannot be factored over the integers. These are called prime or non-factorable polynomials.

• If no pair of factors can be found, the polynomial is prime.

Example: is prime.

Summary Table: Factoring Methods

Practice Exercises

• Factor

• Factor

• Factor

• Factor

• Determine if is factorable

Answers:

• is prime (cannot be factored over the integers)

Additional info: Factoring is essential for solving quadratic equations, simplifying rational expressions, and understanding polynomial functions. Mastery of these techniques is foundational for further study in algebra and calculus.