Problem 4 - Calculus for Biology and Medicine

Question

In Problems , 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 expression for \(f(x)\) is \(x - 4 + \frac{x - 3}{x^2 + x + 3}\).

Step 1: Set Up the Long Division

We will perform long division where the dividend is the polynomial in the numerator, \(x^3 - 3x^2 - 15\), and the divisor is the polynomial in the denominator, \(x^2 + x + 3\). The goal is to express \(f(x)\) as \(Q(x) + \frac{R(x)}{D(x)}\), where \(Q(x)\) is the quotient polynomial and \(R(x)\) is the remainder. Set up the long division with \(x^3 - 3x^2 - 0x - 15\) divided by \(x^2 + x + 3\).

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 \(x\). Place \(x\) above the division bar.

Step 3: Multiply and Subtract

Multiply \(x\) by each term in the divisor, \(x^2 + x + 3\), giving \(x^3 + x^2 + 3x\). Subtract this result from the dividend: \((x^3 - 3x^2 - 0x - 15) - (x^3 + x^2 + 3x) = -4x^2 - 3x - 15\).

Step 4: Repeat Division Steps

Now divide the new leading term \(-4x^2\) by \(x^2\), resulting in \(-4\). Place \(-4\) next to \(x\) in the quotient.

Step 5: Multiply and Subtract Again

Multiply \(-4\) by the divisor \(x^2 + x + 3\), giving \(-4x^2 - 4x - 12\). Subtracting this from \(-4x^2 - 3x - 15\) results in \(x - 3\).

Step 6: Write Final Expression

The quotient is \(x - 4\) and the remainder is \(x - 3\). Thus, \(f(x)\) is expressed as \(x - 4 + \frac{x - 3}{x^2 + x + 3}\).

Key Concepts

Polynomial Division

Polynomial division is like long division with numbers, but it works with polynomials instead. The goal is to divide one polynomial, called the dividend, by another polynomial, the divisor, to obtain a quotient and a remainder. This process can simplify complex expressions and solve polynomial equations more easily.

Here's how it works:

• Set up the division with the dividend and the divisor.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the result by the divisor and subtract from the dividend.
• Repeat the process with the new polynomial until the degree of the remainder is less than the divisor.

In the original exercise, the dividend is a third-degree polynomial, and the divisor is a second-degree polynomial. By performing polynomial division, we simplified the expression into a quotient and a remainder.

Rational Functions

Rational functions are expressions that can be written as the quotient of two polynomials. These functions can often be broken down into simpler components using polynomial division. Understanding this concept ensures you can handle various mathematical problems involving fractions of polynomials.

For instance, the function \( f(x) = \frac{x^3 - 3x^2 - 15}{x^2 + x + 3} \) is a rational function. The process of dividing these polynomials helps in rewriting the function as a sum of a polynomial and a proper rational function. This method deconstructs the complex function into a simpler form, making it easier to analyze or integrate if needed.

During such division, the resulting expression typically takes the form \( Q(x) + \frac{R(x)}{D(x)} \), where \( Q(x) \) is the quotient and \( R(x)/D(x) \) is the simpler rational part.

Quotient and Remainder

The quotient and remainder are crucial aspects of dividing polynomials. When you divide a polynomial by another, you're essentially trying to see how many times the divisor fits into the dividend. The quotient tells you this multiplicative factor, while the remainder indicates what is left after this division process.

In our context, after performing the long division, the quotient is given as \( x - 4 \) and the remainder is \( x - 3 \).

• Quotient \( Q(x) \): The main result after performing the division.
• Remainder \( R(x) \): The leftover part after the division process, smaller than the degree of the divisor.

These elements allow us to write \( f(x) \) as \( x - 4 + \frac{x - 3}{x^2 + x + 3} \). Understanding how to interpret and use the quotient and remainder is vital in simplifying rational functions and solving polynomial equations.