Problem 39 - Algebra and Trigonometry

Question

\(35-82\) Factor the expression completely. $$ x^{2}-2 x-8 $$

Step-by-Step Solution

Answer

The expression factors to \((x - 4)(x + 2)\).

Step 1: Understanding the Problem

The problem asks us to factor the expression \(x^2 - 2x - 8\). Factoring is the process of breaking down an expression into a product of simpler expressions. Our goal is to express the quadratic in the form \((x + a)(x + b)\).

Step 2: Identify Coefficients

First, identify the coefficients of the quadratic expression: \(a = 1\) (the coefficient of \(x^2\)), \(b = -2\) (the coefficient of \(x\)), and \(c = -8\) (the constant term).

Step 3: Finding Two Numbers

Find two numbers that multiply to \(ac\) (\(1 imes -8 = -8\)) and add to \(b\) (\(-2\)). These numbers are \(-4\) and \(2\), since \(-4 imes 2 = -8\) and \(-4 + 2 = -2\).

Step 4: Rewrite and Group Terms

Rewrite the middle term \(-2x\) using the numbers found: \(x^2 - 4x + 2x - 8\). Group the terms into two pairs: \((x^2 - 4x) + (2x - 8)\).

Step 5: Factor by Grouping

Factor out the greatest common factor from each pair: \(x(x - 4) + 2(x - 4)\). Observe that \(x - 4\) is a common factor.

Step 6: Final Solution

Factor out the common term \(x - 4\) to get the final factorized form: \((x - 4)(x + 2)\). This is the completely factored expression.

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two. They are typically written in the standard form as: \[ ax^2 + bx + c \] where:

• \(a\) is the coefficient of the \(x^2\) term,
• \(b\) is the coefficient of the \(x\) term,
• and \(c\) is the constant term, which does not involve \(x\).

Understanding the structure of a quadratic expression is crucial. Each component plays a different role during factoring. In our example, given expression \(x^2 - 2x - 8\), it follows this form with \(a = 1\), \(b = -2\), and \(c = -8\). Recognizing the coefficients helps us proceed to find the factors of the expression.

Factoring by Grouping

Factoring by grouping is an effective method used to factor certain quadratic expressions. This approach involves rearranging and grouping terms so that they can be factored easily. The objective is to look for common factors within each group. For the expression \(x^2 - 2x - 8\), we need to express the middle term in such a way that the expression can be grouped into manageable pairs. We find a pair of numbers whose product is equal to the product of \(a\) and \(c\) (\(-8\)), and whose sum equals \(b\) (\(-2\)). Here, we use \(-4\) and \(2\) because \(-4 \times 2 = -8\) and \(-4 + 2 = -2\). This allows us to rewrite the quadratic expression as \(x^2 - 4x + 2x - 8\). Group and factor each pair: \((x^2 - 4x) + (2x - 8)\). Each group can now be factored separately.

Finding Coefficients

Finding coefficients accurately is the backbone of factoring any quadratic expression. Understanding how to determine these coefficients begins with observation of the quadratic expression in standard form. In \(x^2 - 2x - 8\), the coefficients are straightforward:

• \(a = 1\),
• \(b = -2\),
• and \(c = -8\).

These coefficients guide us in identifying two critical numbers needed for factoring by grouping. The goal is to find numbers whose product is equal to \(a \times c\) and whose sum is \(b\). Thus, understanding, identifying, and utilizing these coefficients correctly leads us directly into effectively dissecting the quadratic into factorable parts.

Factoring Techniques

Several factoring techniques exist for solving quadratic expressions, each suited to varyingly structured problems. It is essential to choose the right one, as this can make the solution process more straightforward and correct. The technique used for \(x^2 - 2x - 8\) is 'factoring by grouping'. Its steps include:

• Identifying two numbers for multiplication and addition based on \(ac\) and \(b\).
• Rewriting the middle term with these numbers to form four terms.
• Grouping into pairs so each can be factored independently.
• Extracting the common factor from both pairs to complete the factorization.

This method uniquely aids in tackling cases where quadratics can be split into simpler linear factors due to its systematic grouping approach.