Problem 13 - Calculus for Biology and Medicine

Question

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{4}+1}{x-1} $$

Step-by-Step Solution

Answer

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

Step 1: Set Up the Division

Write the division problem in long division format. The dividend is \(x^4 + 0x^3 + 0x^2 + 0x + 1\), and the divisor is \(x - 1\). Your initial setup should look like a long division problem with \(x - 1\) outside the division symbol and \(x^4 + 0x^3 + 0x^2 + 0x + 1\) under the division symbol.

Step 2: Divide the First Terms

Divide the first term of the dividend, \(x^4\), by the first term of the divisor, \(x\). This gives \(x^3\). Write \(x^3\) as the first term of the quotient, above the division symbol.

Step 3: Multiply Divisor by the First Term of Quotient

Multiply \(x^3\) by \(x - 1\) to get \(x^4 - x^3\). Write this under the dividend \(x^4 + 0x^3\).

Step 4: Subtract the Result from Dividend

Subtract \(x^4 - x^3\) from \(x^4 + 0x^3\) to find the new dividend: \(x^4 + 0x^3 - (x^4 - x^3) = x^3\).

Step 5: Repeat the Process

Follow the same process: divide \(x^3\) by \(x\) to get \(+x^2\). Write \(x^2\) in the quotient, multiply \(x^2\) by \(x-1\) to get \(x^3 - x^2\), and subtract this from your current remainder \(x^3\) to get a new remainder: \(x^2\).

Step 6: Continue Division

Divide \(x^2\) by \(x\) to get \(x\), write it in the quotient, multiply to get \(x^2 - x\), and subtract from \(x^2\) to get \(x\).

Step 7: Final Division Steps

Divide \(x\) by \(x\) to get \(1\), write it in the quotient, multiply to get \(x - 1\), and subtract from \(x\) to get the remainder \(2\).

Step 8: Form the Final Expression

Now the quotient is \(x^3 + x^2 + x + 1\) and the remainder is \(2\). Therefore, the expression for \(f(x)\) is:\[ f(x) = x^3 + x^2 + x + 1 + \frac{2}{x - 1} \].

Key Concepts

Polynomial Division

Polynomial division is similar to long division with numbers. It is used to divide a polynomial by another polynomial. In our exercise, we used long division to divide the polynomial \(x^4 + 1\) by \(x - 1\). This process involves breaking down the division into manageable steps, where each term of the dividend is divided by the first term of the divisor. This helps in finding the quotient, which is the result of division.
The steps involve writing the polynomials in descending order of their degrees and ensuring each term is accounted for, even if it means using coefficients like 0. Here, we expanded the dividend to \(x^4 + 0x^3 + 0x^2 + 0x + 1\). With each step, you:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the result.
• Subtract this from the current dividend to get a new one.

This process continues until the degree of the new dividend is less than the divisor's degree. The result at this stage is the remainder.

Rational Functions

A rational function is a function that is the ratio of two polynomials. In mathematical terms, it can be expressed as \( f(x) = \frac{p(x)}{q(x)} \) where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) eq 0\). The function we have, \( f(x) = \frac{x^4 + 1}{x-1} \), is an example of a rational function because it is a fraction where the numerator and the denominator are both polynomials.
Rational functions are significant in calculus and algebra for several reasons:

• They often appear in real-world problems involving ratios and rates.
• They help in understanding asymptotic behavior as one examines limits or attempts to find vertical and horizontal asymptotes.
• They are useful in examining piecewise defined functions that include divisions by zero that need careful analysis.

This kind of function extends the concept of ratios from numbers to algebraic expressions, allowing for a rich variety of shapes and behaviors when graphed.

Proper Rational Function

A proper rational function is a special kind of rational function where the degree of the numerator is less than the degree of the denominator. In our example, \( \frac{x^4 + 1}{x-1} \), the division transforms into \( x^3 + x^2 + x + 1 + \frac{2}{x-1} \). Here, \(\frac{2}{x-1}\) is a proper rational function because the degree of \(2\) (i.e., 0) is less than the degree of \(x-1\) (i.e., 1).
Proper rational functions are crucial when simplifying expressions and performing integrations:

• They can be directly integrated using known antiderivatives or partial fraction decomposition.
• They help in analyzing behavior around singularities without the obstruction of higher-degree polynomials in the numerator.
• When addressing improper rational functions, division is used to reformulate them into a polynomial plus a proper rational function.

This breakdown facilitates easier computation and helps in understanding the function's core characteristics, such as continuity and differentiability.