/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 Simplify the expression. $$\fr... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 29

Simplify the expression. $$\frac{x^{2}+1}{x^{2}-4}+\frac{5 x}{x^{2}-4}-\frac{2 x+11}{x^{2}-4}$$

Short Answer

Expert verified
\(\frac{x^{2}+3x-10}{x^{2}-4}\)

Step by step solution

01

Combine Like Terms

Since fractions can be added and subtracted if they have the same denominator, start by combining the numerators: \[(x^{2}+1)+(5x)-(2x+11)\]
02

Simplification

The expression simplifies to: \[x^{2}+5x-2x+1-11\] which further simplifies to: \[x^{2}+3x-10\]. The denominator remains the same.
03

Final Answer

Therefore, the simplified rational expression is \[\frac{x^{2}+3x-10}{x^{2}-4}\].

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Key Concepts

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

Simplifying Expressions
Rational expressions are like any mathematical expressions but written as a fraction where both the numerator and the denominator are polynomials. Simplifying them involves reducing the expression to its simplest form. One way to simplify rational expressions is by combining like terms and performing operations that do not change the value of the expression such as division of numerator and denominator by their greatest common factor. More so, it is important to simplify complex expressions by carrying out operations like addition or subtraction of terms with a common denominator.
Combining Like Terms
In mathematics, combining like terms is essential to simplify expressions. Like terms have the same variables with the same exponents. When you combine them, you add or subtract the coefficients but keep the common variables unchanged. For example, in the expression \(x^2+5x-2x+1-11\), the like terms are \(5x\) and \(-2x\). By combining these terms, you simplify the expression to \(x^2+3x-10\). This step significantly reduces the complexity of the expression and makes further operations easier.
Common Denominator
When working with fractions or rational expressions, having a common denominator is essential for addition or subtraction. This is because you can only add or subtract fractions when they have the same denominator; otherwise, you end up with expressions that are not mathematically equivalent. In the original exercise, all fractions have a denominator of \(x^2-4\), making it simpler to combine the numerators directly. Ensuring a common denominator helps in performing arithmetic operations smoothly and leads to the simplification of complex fractional expressions.
Numerator
The numerator is the top part of a fraction. In a rational expression, it's crucial to focus on simplifying the numerator as this directly influences the overall simplification of the expression. During the simplification process, you perform operations on the numerators while keeping the denominator unchanged until the final expression is obtained. In the exercise, the initial numerators \((x^2+1)+(5x)-(2x+11)\) are simplified to \(x^2+3x-10\). Working on the numerator helps make the fraction more manageable and shapes the rational expression into a form that is easier to understand.

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Most popular questions from this chapter

Sketch the graph of the function. $$y=4 x^{2}-x+6$$

When you add rational expressions, you may need to factor a trinomial to find the LCD. Study the sample below. Then simplify the expressions in Exercises 46鈥�49. $$\text { Sample: } \frac{2 x}{x^{2}-1}+\frac{3}{x^{2}+x-2}=\frac{2 x}{(x+1)(x-1)}+\frac{3}{(x-1)(x+2)}$$ The LCD is \((x+1)(x-1)(x+2)\) Note: If you just used \(\left(x^{2}-1\right)\left(x^{2}+x-2\right)\) as the common denominator, the factor \((x-1)\) would be included twice. $$\frac{5 x-1}{2 x^{2}-7 x-15}-\frac{-3 x+4}{2 x^{2}+5 x+3}$$

Write the equation in standard form. (Lesson 9.5 for 11.7 ) $$9-6 x=2 x^{2}$$

Completely factor the expression. $$6 x^{2}+16 x$$

Simplify the expression. (Review \(8.3 \text { for } 11.7)\) $$\frac{5}{10 x}$$
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