Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=3 x^{3}+2 x^{2}+5 x-2, \quad k=\frac{1}{3}$$

Step-by-Step Solution

Answer

The form of the function in \((x - k) q(x) + r\) will be derived after carrying out polynomial division. Substituting \(k = 1/3\) into the function, the output should equal to the constant term obtained from the polynomial division, which demonstrates \(f(k) = r\).

Step 1: Polynomial Division

First, divide the function \(f(x)=3 x^{3}+2 x^{2}+5 x-2\) by \(x-k\) using polynomial long division, where \(k=1/3\). The long division process gives the quotient function \(q(x)\) and the remainder \(r\).

Step 2: Write the Function in the Required Form

After dividing, write the function in the form \(f(x)=(x-k)q(x)+r\). You should have an equation with \(f(x)\) on one side and \((x-k)q(x) + r\) on the other.

Step 3: Verification

Then, plug the given value \(k=1/3\) into the equation derived in step 2, to demonstrate that \(f(k) = r\). If the left and right hand sides of the equation are the same when evaluated at \(k\), then the transformation and calculations have been done correctly.

Key Concepts

Remainder Theorem

Understanding the Remainder Theorem is crucial for anyone dealing with polynomials. It states that when a polynomial function, let's say \(f(x)\), is divided by a factor of the form \(x-k\), the remainder is simply \(f(k)\).

This theorem is incredibly useful because it allows for a quick evaluation of any polynomial's remainder without having to perform the full division. For instance, in the given exercise, once the function \(f(x) = 3x^3 + 2x^2 + 5x - 2\) has been divided by \(x - k\), and \(k = \frac{1}{3}\), you can immediately find the remainder by calculating \(f\left(\frac{1}{3}\right)\). If done correctly, this evaluated result will be identical to the remainder found using long division.

This can also be flipped around. If you're asked to find \(f(k)\) for some value of \(k\), and you know the remainder when dividing by \(x - k\), you've got your answer without any additional computations. Hence, understanding and applying the Remainder Theorem simplifies polynomial problems significantly.

Synthetic Division

When you're presented with polynomial division, synthetic division serves as a handy shortcut. It's a simpler method compared to the traditional long division and especially useful when dividing by a linear factor.

To perform synthetic division, you take the coefficients of the polynomial and the zero of the divisor. Here's a brief glimpse into the process:

• List out the coefficients of the polynomial.
• Write down the zero of the divisor (for \(x - k\), the zero is \(k\)).
• Follow the synthetic division algorithm to find the new coefficients of the quotient polynomial and the remainder.

This methodology is quicker and less cumbersome, making the division process more approachable and less error-prone for students. For your exercise, the value of \(k = \frac{1}{3}\) could be used in synthetic division to simplify the given polynomial and obtain the quotient and the remainder more efficiently.

Algebraic Functions

When you hear 'algebraic functions,' think about an extensive family of functions you can write using polynomial expressions. Essentially, algebraic functions are built from polynomials using operations add, subtract, multiply, divide, and take roots.

In the exercise, \(f(x)=3x^{3}+2x^{2}+5x-2\) is an example of an algebraic function: a polynomial of degree 3. These functions can be broken down and analyzed, both in their entirety and in parts. By dividing \(f(x)\) by \((x - k)\), and expressing \(f(x)\) as \((x - k)q(x) + r\), you're reformatting the function in a way that highlights the remainder, but the polynomial nature of \(f(x)\) remains.

This reformatting into quotient and remainder is particularly helpful in not only solving equations but also in graphing functions and understanding their properties better. For example, knowing the remainder and the quotient can tell you where the graph of \(f(x)\) intercepts the x-axis or how it behaves around specific values of \(x\).