Problem 9 - Calculus for Biology and Medicine

Question

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}+3 x^{2}+3 x+1}{x^{2}+1} $$

Step-by-Step Solution

Answer

The function \(f(x)\) can be written as \(x + 2 + \frac{3x - 1}{x^{2} + 1}\).

Step 1: Set Up the Long Division

Write the division in the long division format. Place the dividend, \(x^{3} + 3x^{2} + 3x + 1\), inside the division bracket, and the divisor, \(x^{2} + 1\), outside the bracket.

Step 2: Perform Initial Division

Divide the leading term of the dividend, \(x^{3}\), by the leading term of the divisor, \(x^{2}\), to get the first term of the quotient, which is \(x\). Multiply \(x\) by the entire divisor, \(x^{2} + 1\), resulting in \(x^{3} + x\).

Step 3: Subtract and Bring Down Next Term

Subtract \(x^{3} + x\) from the original dividend \(x^{3} + 3x^{2} + 3x + 1\). This gives \(2x^{2} + 3x + 1\) as the new dividend. Bring down the remainder, which is \(+1\), to form \(2x^{2} + 3x + 1\).

Step 4: Repeat Division Process

Divide the new leading term \(2x^{2}\) by \(x^{2}\) to get \(2\). Multiply \(2\) by the divisor \(x^{2} + 1\), obtaining \(2x^{2} + 2\).

Step 5: Subtract Again

Subtract \(2x^{2} + 2\) from \(2x^{2} + 3x + 1\). This results in a remainder of \(3x - 1\).

Step 6: Express the Result

The solution to the long division is \(x + 2\) with a remainder of \(3x - 1\). Therefore, the expression is written as \(x + 2 + \frac{3x - 1}{x^{2} + 1}\).

Key Concepts

Polynomial Division

Polynomial division is similar to regular long division that you might know from arithmetic with numbers, but it involves polynomials instead of digits. The goal is to divide one polynomial (the dividend) by another polynomial (the divisor) to get a quotient and a remainder. In this context, it helps to simplify complex polynomial expressions by breaking them into more manageable parts.

When you perform polynomial division, you start by dividing the leading term (the term with the highest power) of the dividend by the leading term of the divisor. Here, you are essentially seeing how many times the divisor can "fit" into the dividend. The result gives you the first term of the quotient.

• Set up the division problem by writing down your polynomials: the dividend inside the division bracket and the divisor outside.
• Divide the leading terms to find the first term of the quotient.
• Multiply the entire divisor by this quotient term, and subtract the result from the original dividend.
• Bring down the next term of the dividend and repeat the process until you've addressed all the terms.

Through these steps, you effectively dissect the polynomial into a quotient and a remainder that cannot be further divided by the divisor.

Rational Functions

Rational functions are expressions that involve the division of two polynomials. The function takes the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions appear often in calculus and algebra, and they can behave quite differently depending on their composition.

In the context of this exercise, after performing polynomial long division, the function \( f(x) \) was rewritten as a sum of a polynomial and a proper rational function (a rational function where the degree of the numerator is less than the degree of the denominator). This proper rational function is crucial because it often gives insights into the behavior of the function, such as vertical asymptotes or points where the function is undefined.

Thus, by dividing polynomials and rewriting rational functions in this manner, you simplify the expressions, making it easier to understand and analyze the characteristics of these complex functions in mathematical analysis.

Calculus Techniques

Calculus techniques often involve working with polynomials and rational functions extensively, especially when dealing with limits, derivatives, and integrals. Long division helps precursors to these techniques, such as simplifying expressions and identifying limits.

For example, knowing how to use polynomial division allows you to tackle integrals involving rational functions more efficiently. You simplify the function first and then apply integration techniques without dealing with a complex polynomial in the numerator directly.

• Understand how simplified expressions can make differentiation and integration more straightforward.
• Identify asymptotic behavior more easily, as simplifying rational functions can make it more transparent.
• Examine limit values of functions at infinity, as breaking down rational functions simplifies determining the limit behavior as well.

These processes in calculus are greatly facilitated by techniques like long division for polynomials, ensuring a smoother path to finding solutions in calculus problems.