Problem 93 - Elementary Algebra

Question

Find the quotient. \(\frac{x^{2}-1}{4 x^{2}-1} \div \frac{x-1}{2 x+1}\).

Step-by-Step Solution

Answer

Question: Find the quotient of the given fractions, \(\frac{x^{2}-1}{4 x^{2}-1} \div \frac{x-1}{2 x+1}\). Answer: The simplified quotient of the given expression is \(\frac{x+1}{2x-1}\).

Step 1: Factorize expressions where possible

First, we will factorize the expressions given in the problem. The expressions are: 1. The numerator of the first fraction: \(x^2 - 1\) 2. The denominator of the first fraction: \(4x^2 - 1\) 3. The numerator of the second fraction: \(x - 1\) 4. The denominator of the second fraction: \(2x + 1\) 1. The expression \(x^2 - 1\) can be factorized using difference of squares, which results in \((x+1)(x-1)\). 2. The expression \(4x^2 - 1\) can be factorized using difference of squares, which results in \((2x+1)(2x-1)\). 3. The expression \(x-1\) is already in its simplest form. 4. The expression \(2x+1\) is already in its simplest form.

Step 2: Rewrite the problem as a multiplication

We are given the problem \(\frac{x^{2}-1}{4 x^{2}-1} \div \frac{x-1}{2 x+1}\). To perform the division, we can rewrite it as a multiplication by multiplying the first fraction by the reciprocal of the second fraction: \(\frac{x^{2}-1}{4 x^{2}-1} \times \frac{2 x+1}{x-1}\) Now, we can substitute the factorized expressions we found in Step 1: \(\frac{(x+1)(x-1)}{(2x+1)(2x-1)} \times \frac{2 x+1}{x-1}\)

Step 3: Simplify the expression

To simplify the expression, we can cancel out common factors from the numerator and the denominator: \(\frac{(x+1)\cancel{(x-1)}}{(\cancel{(2x+1)})(2x-1)} \times \frac{\cancel{2 x+1}}{\cancel{x-1}}\) After canceling the common factors, we are left with: \(\frac{(x+1)}{(2x-1)}\)

Step 4: Write the final answer

The simplified quotient of the given expression is: \(\frac{x+1}{2x-1}\)

Key Concepts

Algebraic Division

Algebraic division, sometimes known as polynomial division, is a method used to divide one algebraic expression by another, similar to long division with numbers. In our example, we are tasked with dividing two rational expressions. The process involves using the factoring of polynomials to simplify expressions before division.

When presented with a division problem like \(\frac{x^{2}-1}{4 x^{2}-1} \div \frac{x-1}{2 x+1}\), the first move is to change it into a multiplication problem. We do this by multiplying the first expression by the reciprocal of the second expression, as we can see from the textbook's step-by-step solution. This approach transforms the problem into a much simpler task that involves multiplication and enables the cancellation of like terms, which is a core part of simplifying expressions.

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra. It involves reducing an algebraic expression to its simplest form so that it is easier to understand or further manipulate. This is done by combining like terms, factoring, and canceling out terms where possible.

In the case of our original division problem, simplification comes into play after converting the division into multiplication and factoring the polynomials. As shown in the solution, simplifying the expression involves identifying and canceling factors that appear in both the numerator and denominator. It streamlines the process and leads us to the most reduced form of the expression, which is \(\frac{x+1}{2x-1}\). By simplifying, we move closer to the solution and also get a clearer picture of the relationships between the algebraic terms.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components ('factors') that, when multiplied together, give back the original polynomial. This is especially useful when working with division problems in algebra, as it often reveals common factors that can be canceled to simplify the expression.

In our division problem, both \(x^2 - 1\) and \(4x^2 - 1\) are instances of a 'difference of squares'. They can be factored into \(x+1)(x-1)\) and \(2x+1)(2x-1)\), respectively. Recognizing these patterns is crucial to simplifying the given expressions and is a testament to the importance of understanding how to factor polynomials effectively. Factoring enables the cancelation process in simplification and is a stepping-stone to revealing the quotient of a division problem in algebra.