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Chapter 7: Problem 65

Divide and, if possible, simplify. $$(2 x-1) \div \frac{2 x^{2}-11 x+5}{4 x^{2}-1}$$

Short Answer

Expert verified
\frac{2x + 1}{x - 5}

Step by step solution

01

Rewrite the Division as Multiplication

To divide by a fraction, multiply by its reciprocal. Rewrite the given expression \( (2x - 1) \div \frac{2x^2 - 11x + 5}{4x^2 - 1} \) into \( (2x - 1) \times \frac{4x^2 - 1}{2x^2 - 11x + 5} \).
02

Factor the Numerator and Denominator

Factor the quadratic expressions in the numerator and the denominator:1. \( 4x^2 - 1 \) is a difference of squares and can be factored as \( (2x + 1)(2x - 1) \).2. \( 2x^2 - 11x + 5 \) can be factored as \( (2x - 1)(x - 5) \).
03

Substitute the Factors Back into the Expression

Rewrite the expression using the factored forms: \( (2x - 1) \times \frac{(2x + 1)(2x - 1)}{(2x - 1)(x - 5)} \).
04

Cancel Common Factors

Cancel the common factor \( (2x - 1) \) in the numerator and the denominator. The expression reduces to \( (2x + 1) \div (x - 5) \).
05

Write the Simplified Expression

After cancelling the common factor, the simplified form of the expression is \( \frac{2x + 1}{x - 5} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Division
Polynomial division involves dividing one polynomial by another. In our example, we start with the expression \( (2x - 1) \div \frac{2x^2 - 11x + 5}{4x^2 - 1} \).When dividing by a fraction, we multiply by its reciprocal. Thus, the division becomes a multiplication:\( (2x - 1) \times \frac{(4x^2 - 1)}{(2x^2 - 11x + 5)} \).This is the first step in polynomial division. Now, our goal is to simplify the resulting expression.
Factoring Quadratic Expressions
Next, we need to factor the quadratic expressions in the numerator and the denominator. Factoring is the process of breaking down expressions into their simpler components. Here, we have two factorizations:
  • For \( 4x^2 - 1 \), we recognize it as a difference of squares, which can be factored to \( (2x + 1)(2x - 1) \).
  • For \( 2x^2 - 11x + 5 \), we look for two numbers that multiply to give \( 2 * 5 = 10 \) and add up to \( -11 \). These numbers are \( -1 \) and \( -10 \).Thus, it factors into \( (2x - 1)(x - 5) \).
After factoring, we substitute the factors back into the expression, making it easier to simplify.
Simplifying Rational Expressions
Simplifying rational expressions involves reducing the expression to its simplest form. Here are the steps to simplify the expression given in the exercise:
  • Substitute back the factored forms to get \( (2x - 1) \times \frac{(2x + 1)(2x - 1)}{(2x - 1)(x - 5)} \).
  • Cancel out the common factors, which in this case is \( (2x - 1) \).
  • The expression simplifies down to \( \frac{(2x + 1)}{(x - 5)} \).
This final answer, \( \frac{2x + 1}{x - 5} \), is the simplified form of the initial expression. By following these steps, you can simplify any similar rational expression efficiently.

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Most popular questions from this chapter

For each pair of functions fand \(g,\) find all values of a for which \(f(a)=g(a)\). $$f(x)=\frac{x+3}{x+2}-\frac{x+4}{x+3}, g(x)=\frac{x+5}{x+4}-\frac{x+6}{x+5}$$

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