Express Your Answer In The Form $p(x)+\frac{k}{x}$ Where $p(x)$ Is A Polynomial And \$k$[/tex\] Is An Integer.$\frac{x^4+2x^2-5}{x}=\square$

Introduction

When dealing with rational functions, it's often necessary to express them in a specific form to simplify or analyze their behavior. In this case, we're tasked with expressing the rational function $\frac{x^4+2x^2-5}{x}$ in the form $p(x)+\frac{k}{x}$, where $p(x)$ is a polynomial and $k$ is an integer. This form is particularly useful for understanding the behavior of the function as $x$ approaches zero.

Understanding the Given Rational Function

The given rational function is $\frac{x^4+2x^2-5}{x}$. To express this function in the desired form, we need to manipulate the numerator to separate it into a polynomial and a fraction with a denominator of $x$. This can be achieved by factoring the numerator and then rearranging the terms.

Factoring the Numerator

The numerator of the given rational function is $x^4+2x^2-5$. We can factor this expression by recognizing that it can be written as a difference of squares:

$x^4+2x^2-5 = (x^2+5)(x^2-1)$

Rearranging the Terms

Now that we have factored the numerator, we can rearrange the terms to separate the polynomial and the fraction with a denominator of $x$. We can do this by multiplying the fraction by $\frac{x}{x}$, which is equivalent to 1:

$\frac{x^4+2x^2-5}{x} = \frac{(x^2+5)(x^2-1)}{x} = \frac{(x^2+5)x^2}{x} - \frac{(x^2-1)}{x}$

Simplifying the Expression

Now that we have separated the polynomial and the fraction with a denominator of $x$, we can simplify the expression further. We can do this by canceling out the common factor of $x$ in the numerator and denominator of the first term:

$\frac{(x^2+5)x^2}{x} - \frac{(x^2-1)}{x} = (x^2+5)x - \frac{(x^2-1)}{x}$

Expressing the Rational Function in the Desired Form

Now that we have simplified the expression, we can express the rational function in the desired form. We can do this by combining the polynomial and the fraction with a denominator of $x$:

$(x^2+5)x - \frac{(x^2-1)}{x} = x^3+5x - \frac{(x^2-1)}{x}$

Identifying the Polynomial and the Integer

Now that we have expressed the rational function in the desired form, we can identify the polynomial and the integer. The polynomial is $p(x) = x^3+5x$, and the integer is $k = -(x^2-1)$.

Conclusion

In this article, we have expressed the rational function $\frac{x^4+2x^2-5}{x}$ in the form $p(x)+\frac{k}{x}$, where $p(x)$ is a polynomial and $k$ is an integer. We have achieved this by factoring the numerator, rearranging the terms, simplifying the expression, and combining the polynomial and the fraction with a denominator of $x$. This form is particularly useful for understanding the behavior of the function as $x$ approaches zero.

Final Answer

The final answer is $\boxed{x^3+5x-\frac{x^2-1}{x}}$.

Additional Information

• The polynomial $p(x)$ is $x^3+5x$.
• The integer $k$ is $-(x^2-1)$.
• The rational function can be expressed in the desired form by factoring the numerator, rearranging the terms, simplifying the expression, and combining the polynomial and the fraction with a denominator of $x$.

Related Topics

• Rational functions
• Polynomials
• Factoring
• Simplifying expressions
• Combining terms

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Rational Functions" by Wolfram MathWorld
  

Introduction

In our previous article, we explored how to express a rational function in the form $p(x)+\frac{k}{x}$, where $p(x)$ is a polynomial and $k$ is an integer. In this article, we'll answer some common questions related to this topic.

Q: What is the purpose of expressing a rational function in this specific form?

A: Expressing a rational function in the form $p(x)+\frac{k}{x}$ is useful for understanding the behavior of the function as $x$ approaches zero. This form helps to identify the polynomial and the integer, which can be useful for simplifying or analyzing the function.

Q: How do I factor the numerator of a rational function?

A: Factoring the numerator of a rational function involves recognizing common factors or using techniques such as the difference of squares. For example, if the numerator is $x^4+2x^2-5$, we can factor it as $(x^2+5)(x^2-1)$.

Q: What if the numerator cannot be factored?

A: If the numerator cannot be factored, we can still express the rational function in the desired form by rearranging the terms and simplifying the expression. We can do this by multiplying the fraction by $\frac{x}{x}$, which is equivalent to 1.

Q: How do I identify the polynomial and the integer in the desired form?

A: To identify the polynomial and the integer, we need to look at the expression in the desired form. The polynomial is the part of the expression that does not have a denominator of $x$, and the integer is the constant term in the expression.

Q: What if the integer is not an integer?

A: If the integer is not an integer, we can still express the rational function in the desired form. However, the integer will be a rational number, not an integer.

Q: Can I use this technique to express any rational function in the desired form?

A: Yes, this technique can be used to express any rational function in the desired form. However, the process may be more complicated for some functions, and it may be necessary to use additional techniques or tools.

Q: Are there any limitations to this technique?

A: Yes, there are limitations to this technique. For example, it may not be possible to express a rational function in the desired form if the numerator and denominator have no common factors.

Q: Can I use this technique to solve problems in calculus?

A: Yes, this technique can be used to solve problems in calculus. For example, it can be used to find the limit of a rational function as $x$ approaches zero.

Q: Are there any other applications of this technique?

A: Yes, there are other applications of this technique. For example, it can be used to simplify or analyze rational functions in other areas of mathematics, such as algebra and number theory.

Conclusion

In this article, we've answered some common questions related to expressing rational functions in the form $p(x)+\frac{k}{x}$. We've discussed the purpose of this technique, how to factor the numerator, how to identify the polynomial and the integer, and some limitations of the technique. We've also explored some applications of this technique in calculus and other areas of mathematics.

Final Answer

The final answer is $\boxed{yes}$.

Additional Information

• The technique of expressing a rational function in the form $p(x)+\frac{k}{x}$ is useful for understanding the behavior of the function as $x$ approaches zero.
• Factoring the numerator involves recognizing common factors or using techniques such as the difference of squares.
• The polynomial is the part of the expression that does not have a denominator of $x$, and the integer is the constant term in the expression.
• This technique can be used to solve problems in calculus and other areas of mathematics.

Related Topics

• Rational functions
• Polynomials
• Factoring
• Simplifying expressions
• Combining terms

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Rational Functions" by Wolfram MathWorld