Problem 14 - Calculus for Biology and Medicine

Question

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{5}-1}{x-1} $$

Step-by-Step Solution

Answer

The quotient is \( x^4 + x^3 + x^2 + x + 1 \).

Step 1: Set Up the Division

In polynomial long division, arrange the dividend and divisor. The dividend is the polynomial in the numerator \( x^5 - 1 \) and the divisor is \( x - 1 \). The division setup resembles that of numerical long division.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \( x^5 \) by the leading term of the divisor \( x \). This gives \( x^4 \). Write \( x^4 \) above the long division bar.

Step 3: Multiply and Subtract

Multiply \( x^4 \) by the divisor \( x - 1 \) to get \( x^5 - x^4 \). Subtract \( x^5 - x^4 \) from \( x^5 - 1 \) to get \( x^4 - 1 \). Carry this result down.

Step 4: Repeat Division

Divide the new leading term \( x^4 \) by \( x \) to get \( x^3 \). Repeat the process: multiply \( x^3 \) by \( x - 1 \) to get \( x^4 - x^3 \), and subtract from \( x^4 - 1 \) to get \( x^3 - 1 \).

Step 5: Continue the Process

Continue this process for each successive leading term: Divide \( x^3 \) by \( x \), multiply, subtract, and bring down the next term. You get \( x^2 \), then \( x \), and finally \( 1 \), leading to zero remainder.

Step 6: Verify and Write the Result

The division is complete when the remainder is zero. Our quotient is \( x^4 + x^3 + x^2 + x + 1 \). This represents the polynomial part of \( f(x) \). Since there's no remainder, the proper rational function is zero.

Key Concepts

Understanding Rational Functions

Rational functions are expressions where a polynomial is divided by another polynomial.
These expressions are similar to fractions, but instead of numbers in the numerator and the denominator, we have polynomials.
For example, in the exercise above, we have the rational function \( \frac{x^5 - 1}{x - 1} \). In this function, the numerator \( x^5 - 1 \) is one polynomial, and the denominator \( x - 1 \) is another. When the degree of the numerator is greater than the degree of the denominator, the function is considered improper.
The task is often to transform it into a sum of a polynomial and a proper rational function. Proper rational functions have numerators with a lower degree than their denominators, making them easier to analyze especially in calculus.

The Process of Long Division for Polynomials

Long division is not only for numbers! In math, we can extend it to polynomials, making it a useful tool in algebra and calculus.
The process of polynomial long division mirrors the one we use in numerical division, but with extra steps to handle variables. To perform long division on a polynomial:

• Arrange your terms in descending order of degree, if they're not already.
• Divide the leading term of your dividend by the leading term of your divisor to find the first term of the quotient.
• Multiply your entire divisor by this new term of the quotient, and subtract it from the original dividend.
• Repeat the process with what remains until the degree of the remainder is less than the degree of the divisor.

In the provided solution, we see this process executed as we reduce \( x^5 - 1 \) using \( x - 1 \). After reducing step by step, we discovered that the remainder was zero, leading us to the final polynomial expression, confirming the proper rational function part is zero.

How Calculus Relates to Polynomial Division

Calculus often requires simplifying expressions, a task where polynomial division becomes highly beneficial.
Many calculus problems involve integration or differentiation of rational functions, especially improper ones. Improper rational functions can be challenging to deal with directly.
By using polynomial long division, we convert them into sums of polynomials and proper rational functions, which are much easier to integrate or differentiate.

• When finding antiderivatives, polynomials result in simpler expressions post-integration.
• Derivative calculations become straightforward, allowing for easy application of basic calculus rules.

Thus, mastering polynomial division strengthens one's ability to work through calculus problems efficiently, making the process smoother and results more digestible in further mathematical applications.