Problem 61 - A Graphical Approach to College Algebra

Question

Factor, using the given common factor. Assume that all variables represent positive real numbers. $$4 k^{-1}+k^{-2} ; \quad k^{-2}$$

Step-by-Step Solution

Answer

The expression \(4k^{-1} + k^{-2}\) factors to \(k^{-2}(4k + 1)\).

Step 1: Identify the Common Factor

Examine the expression that needs factoring which is \(4k^{-1} + k^{-2}\). The given common factor is \(k^{-2}\). Both terms in the expression have powers of \(k\) where \(k^{-2}\) is part of each term.

Step 2: Factor Out the Common Term

Divide each term by the common factor \(k^{-2}\). For \(4k^{-1}\), divide by \(k^{-2}\):\[\frac{4k^{-1}}{k^{-2}} = 4k\]For \(k^{-2}\), divide by \(k^{-2}\):\[\frac{k^{-2}}{k^{-2}} = 1\]Thus, the expression becomes: \[k^{-2}(4k + 1)\].

Step 3: Verify the Factoring

Expand the factored form \(k^{-2}(4k + 1)\) to check:\[k^{-2} imes 4k + k^{-2} imes 1 = 4k^{-1} + k^{-2}\]. Both expressions match, confirming our factorization is correct.

Key Concepts

Common Factor

A common factor is a term that divides all components of an algebraic expression without leaving a remainder. It's fundamental in simplifying and solving polynomial equations. In our exercise, we need to factor using the common factor \(k^{-2}\). Let's break it down simply:

• First, identify if a common factor is present in the terms.
• Ensure this factor can divide each term evenly.
• In the expression \(4k^{-1} + k^{-2}\), both terms include \(k^{-2}\) as a common factor.

Factoring out a common factor helps in simplifying expressions, making it easier to further solve or integrate them into larger, complex equations. It's like reducing fractions to their simplest form.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the building blocks of algebra. Here are a few things to remember:

• They can be simple, like \(x + 2\), or complex, like \(4k^{-1} + k^{-2}\).
• Algebraic expressions are used to represent real-world situations and solve mathematical problems.
• When factoring expressions, the goal is often to simplify or prepare them for solving equations.

In the context of the exercise, recognizing the structure of the algebraic expression \(4k^{-1} + k^{-2}\) helps us identify opportunities to use factorization techniques effectively.

Real Numbers

Real numbers include all the numbers you can find on the number line. These are every number that isn’t imaginary or complex. They consist of:

• Positive and negative numbers.
• Whole numbers, continuations like decimals, and fractions.
• Both rational and irrational numbers.

In our exercise, all variables assume positive real numbers. This assumption simplifies the factorization process since negative or zero values could fundamentally change the interpretation and result of the expressions. Understanding that the given variables represent positive real numbers ensures we're working within a typical, concrete arithmetic framework.