Problem 6 - Calculus for Biology and Medicine

Question

Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}-3 x^{2}-15}{x^{2}+x+3} $$

Step-by-Step Solution

Answer

The function is expressed as: \[ f(x) = x - 4 + \frac{x - 3}{x^2 + x + 3} \]

Step 1: Set up the division

Set up the long division problem by placing the dividend, \(x^3 - 3x^2 - 15\), under the division bar and the divisor, \(x^2 + x + 3\), outside it. This will allow us to perform long division to simplify the function.

Step 2: Divide the leading terms

Divide the leading term of the dividend, \(x^3\), by the leading term of the divisor, \(x^2\). This gives us \(x\). Write \(x\) above the division bar.

Step 3: Multiply and subtract

Multiply \(x\) by the entire divisor \((x^2 + x + 3)\) to get \(x^3 + x^2 + 3x\). Subtract this result from the original dividend \((x^3 - 3x^2 - 15)\) to find the new dividend, which is \(-4x^2 - 3x - 15\).

Step 4: Repeat the division step

Divide the new leading term \(-4x^2\) by the leading term of the divisor, \(x^2\). This gives \(-4\). Write \(-4\) above the division bar next to the \(x\).

Step 5: Multiply and subtract again

Multiply \(-4\) by the entire divisor \((x^2 + x + 3)\) to get \(-4x^2 - 4x - 12\). Subtract this from the current dividend \(-4x^2 - 3x - 15\) to find the remainder, which is \(x - 3\).

Step 6: Express the result

Since \(x^2 + x + 3\) cannot further divide \(x - 3\), the division process is complete. The quotient is \(x - 4\) and the remainder is \(x - 3\). Therefore, the function can be expressed as a sum of a polynomial and a proper rational function: \[ f(x) = x - 4 + \frac{x - 3}{x^2 + x + 3} \]

Key Concepts

Proper Rational Function

In algebra, a proper rational function is a fraction where the degree of the numerator is less than the degree of the denominator. When expressing a function like \(f(x) = \frac{x^3 - 3x^2 - 15}{x^2 + x + 3}\), the goal is often to simplify it into a form where a polynomial plus a proper rational function is present. This separation helps in recognizing the different parts that contribute to the behavior of the function, especially when graphing or integrating.

For instance, after performing the polynomial long division on \(f(x)\), we end up with \(f(x) = x - 4 + \frac{x - 3}{x^2 + x + 3}\). Here, \(x - 4\) is the polynomial part, and \(\frac{x - 3}{x^2 + x + 3}\) is the proper rational function since \(\deg(x - 3) = 1\) is less than \(\deg(x^2 + x + 3) = 2\).

• Numerator degree: Lower than the denominator.
• Helps in decomposition and analysis of complex functions.

Understanding this distinction is crucial in various calculus and algebra problems where determining limits and understanding asymptotes play a key role.

Remainder Theorem

The Remainder Theorem provides us with a valuable tool to determine the remainder of a polynomial division without performing the full division process. Specifically, for a polynomial \(P(x)\), if it is divided by a binomial of the form \(x - c\), the remainder of this division is simply \(P(c)\).

However, when dealing with polynomial long division for functions like \(f(x) = \frac{x^3 - 3x^2 - 15}{x^2 + x + 3}\), the Remainder Theorem directly might not apply because it involves divisors more complex than \(x - c\). Instead, long division is necessary to find the remainder, which represents the leftover part that cannot be further divided by the given divisor.

• Allows quick computation of remainder for simple divisors \(x - c\).
• Illustrates connection between polynomial roots and division.

In the context of our example, the remainder \(x - 3\) was determined through stepwise division. It shows what remains of the function \(f(x)\) after the leading terms have been addressed through polynomial subtraction.

Dividend and Divisor in Algebra

When performing polynomial division, it's key to identify the dividend and divisor correctly:

• **Dividend**: The polynomial you want to divide, in this case, \(x^3 - 3x^2 - 15\).
• **Divisor**: The polynomial you are dividing by, which is \(x^2 + x + 3\) in our example.

The process involves arranging these polynomials in long division format, similar to arithmetic division. Place the dividend under the division bar and the divisor outside. This setting enables us to systematically find the quotient and remainder.

Understanding the roles of dividend and divisor is crucial. It ensures each step of division follows logically, allowing for accurate results. Proper setup helps avoid errors in calculation and interpretation, particularly when these expressions become parts of larger algebraic manipulation tasks. Every detail matters, from ensuring the highest degree terms of both expression line up to managing the step-by-step subtraction carefully.