Problem 23 - Elementary and Intermediate Algebra

Question

a. Write \(5 x^{2}+35 x\) as a fraction. b. What is the reciprocal of \(5 x^{2}+35 x ?\)

Step-by-Step Solution

Answer

\(5x^2 + 35x\) as a fraction is \(\frac{5x(x + 7)}{1}\) and its reciprocal is \(\frac{1}{5x(x + 7)}\).

Step 1: Factor the Expression

First, we need to factor the expression \(5x^2 + 35x\). We see that both terms, \(5x^2\) and \(35x\), have a common factor. The greatest common factor (GCF) of the numerical coefficients 5 and 35 is 5, and there is an \(x\) common to both terms. This gives us a common factor of \(5x\). We factor out \(5x\) from each term: \(5x(x + 7)\).

Step 2: Express as a Fraction

To write \(5x^2 + 35x\) as a fraction, consider the expression we obtained from factoring: \(5x(x + 7)\). We can express any expression as a fraction by putting it over 1. Therefore, \(5x^2 + 35x\) expressed as a fraction is \(\frac{5x(x + 7)}{1}\).

Step 3: Find the Reciprocal

The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). Applying this to \(\frac{5x(x + 7)}{1}\), the reciprocal becomes \(\frac{1}{5x(x + 7)}\).

Key Concepts

Greatest Common Factor (GCF)

In algebra, factoring expressions is a key element in simplifying and solving problems. One of the essential steps is finding the Greatest Common Factor (GCF). The GCF is the largest number or expression that divides every term without leaving a remainder. In the expression \(5x^2 + 35x\), we need to identify a factor common to both terms: \(5x^2\) and \(35x\).

Let's break this down:

• For the numbers 5 and 35, the greatest common factor is 5 because it is the largest number that divides both 5 and 35.
• When it comes to the variable part, each term has at least one \(x\). Thus, \(x\) is the common variable factor.

Therefore, the GCF of \(5x^2 + 35x\) is \(5x\). By factoring out \(5x\) from the expression, we rewrite it as \(5x(x + 7)\), effectively simplifying it for further operations.

Reciprocal of a Polynomial

Understanding the reciprocal of a polynomial involves transforming the expression into a fraction. The reciprocal is simply the inverse of that fraction. Suppose we have an expression represented in the fraction form \(\frac{a}{b}\).

The reciprocal is \(\frac{b}{a}\). For expressions such as \(5x^2 + 35x\), let's first factor the polynomial, where we get \(5x(x + 7)\).

When expressed as a fraction, it becomes \(\frac{5x(x + 7)}{1}\). Here's how you find the reciprocal:

• Swap the numerator and the denominator to transform the expression \(\frac{5x(x + 7)}{1}\) into \(\frac{1}{5x(x + 7)}\).

This reciprocal form demonstrates how polynomials can be inverted, which is particularly useful in algebraic equations where division by a polynomial is involved.

Expression as a Fraction

Writing algebraic expressions as fractions is another fundamental concept in algebra that aids in simplification and problem-solving. A fraction essentially represents a division. Any polynomial or expression can technically be turned into a fraction by placing it over 1.

Consider the expression \(5x^2 + 35x\). After factoring this into \(5x(x + 7)\), we can easily write it in the fraction form:

\[ \frac{5x(x + 7)}{1} \]

This formulation doesn't change the value of the expression but gives it a structural form that is often easier to manipulate in equations, especially when taking reciprocals or performing algebraic divisions.