The Given Polynomial Expression

Oct 13, 2025 by pynkocean.com 1 views

===========================================================

Introduction

Factoring polynomial expressions is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. In this article, we will focus on factoring a given polynomial expression and explore the different methods used to achieve this. We will also discuss the importance of factoring in mathematics and its applications in various fields.

What is Factoring?

Factoring is the process of expressing a polynomial expression as a product of simpler expressions, called factors. These factors can be numbers, variables, or a combination of both. Factoring helps us to simplify complex expressions, identify common factors, and solve equations.

The Given Polynomial Expression

The given polynomial expression is:

${15x^2 - 2x - 8}$

Our goal is to factor this expression into a product of simpler expressions.

Method 1: Factoring by Grouping

One method of factoring is by grouping. This method involves grouping the terms of the polynomial expression into pairs and then factoring out the common factors from each pair.

Step 1: Group the Terms

Group the terms of the polynomial expression into pairs:

${15x^2 - 2x - 8 = (15x^2 - 8) - 2x}$

Step 2: Factor Out the Common Factors

Factor out the common factors from each pair:

${(15x^2 - 8) - 2x = 4(3x^2 - 2) - 2x}$

Step 3: Simplify the Expression

Simplify the expression by combining like terms:

${4(3x^2 - 2) - 2x = 4(3x^2 - 2) - 2x = 12x^2 - 8 - 2x}$

Step 4: Factor the Expression

Factor the expression by grouping:

${12x^2 - 8 - 2x = (12x^2 - 2x) - 8 = 2x(6x - 1) - 8}$

Step 5: Factor Out the Common Factor

Factor out the common factor from the expression:

${2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 =

===========================================================

Introduction

What is Factoring?

The Given Polynomial Expression

The given polynomial expression is:

[15x^2 - 2x - 8}$

Our goal is to factor this expression into a product of simpler expressions.

Method 1: Factoring by Grouping

One method of factoring is by grouping. This method involves grouping the terms of the polynomial expression into pairs and then factoring out the common factors from each pair.

Step 1: Group the Terms

Group the terms of the polynomial expression into pairs:

${15x^2 - 2x - 8 = (15x^2 - 8) - 2x}$

Step 2: Factor Out the Common Factors

Factor out the common factors from each pair:

${(15x^2 - 8) - 2x = 4(3x^2 - 2) - 2x}$

Step 3: Simplify the Expression

Simplify the expression by combining like terms:

${4(3x^2 - 2) - 2x = 4(3x^2 - 2) - 2x = 12x^2 - 8 - 2x}$

Step 4: Factor the Expression

Factor the expression by grouping:

${12x^2 - 8 - 2x = (12x^2 - 2x) - 8 = 2x(6x - 1) - 8}$

Step 5: Factor Out the Common Factor

Factor out the common factor from the expression:

[2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 = 2x(6x - 1) - 8 =