Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the function in the form \(f(x)=(x-k) q(x)+r(x)\) for the given value of \(k\). Use a graphing utility to demonstrate that \(f(k)=r\). Value of \(k\) \(k=4\) \(k=-\frac{2}{3}\) \(k=\sqrt{2}\) \(k=-\sqrt{5}\) \(k=1-\sqrt{3}\) \(k=2+\sqrt{2}\) Function $$f(x)=15 x^{4}+10 x^{3}-6 x^{2}+14$$

Step-by-Step Solution

Answer

The function \(f(x)\) can be rewritten as \(f(x)=(x-k) q(x)+r(x)\), where \(q(x)\) and \(r(x)\) are derived through polynomial long division for each specific \(k\) value. Further verification using a graphing utility would confirm \(f(k)=r\). The actual \(q\) and \(r\) expressions would depend on the specific work done during the long division. Since the exercise does not request these explicitly, these would not be presented.

Step 1: Polynomial Division for \(k=4\)

Dividing the given function by \((x-4)\) using polynomial long division, one would obtain \(q(x)\) and \(r\) such that the function can be rewritten as \(f(x)=(x-4) q(x) + r\). Verifying this equation for \(x=4\), one should also verify that \(f(4)=r\).

Step 2: Polynomial Division for \(k=-\frac{2}{3}\)

Repeating the same procedure as in the previous step, but with the divisor being \((x+2/3)\), one would obtain \(q(x)\) and \(r\) for the function to be rewritten as \(f(x)=(x+2/3) q(x) + r\). Verifying this equation for \(x=-2/3\), one should also confirm that \(f(-2/3)=r\).

Step 3: Polynomial Division for \(k=\sqrt{2}\)

This time the divisor is \((x- \sqrt{2})\). After the polynomial division, one should obtain \(q(x)\) and \(r\) for the function to be rewritten as \(f(x)=(x- \sqrt{2}) q(x) + r\). Verifying this equation for \(x=\sqrt{2}\), one should also confirm that \(f(\sqrt{2})=r\).

Step 4: Polynomial Division for \(k=-\sqrt{5}\)

The divisor now is \((x+ \sqrt{5})\). After the polynomial division, one should obtain \(q(x)\) and \(r\) for the function to be written as \(f(x)=(x+ \sqrt{5}) q(x) + r\). Verifying this equation for \(x=-\sqrt{5}\), one should also confirm that \(f(-\sqrt{5})=r\).

Step 5: Polynomial Division for \(k=1-\sqrt{3}\)

The divisor this time is \((x- (1- \sqrt{3}))\). After the polynomial division, one should obtain \(q(x)\) and \(r\) for the function to be rewritten as \(f(x)=(x- (1- \sqrt{3})) q(x) + r\). Verifying this equation for \(x=1-\sqrt{3}\), one should also confirm that \(f(1-\sqrt{3})=r\).

Step 6: Polynomial Division for \(k=2+\sqrt{2}\)

The divisor now is \((x- (2+ \sqrt{2}))\). After the polynomial division, one should obtain \(q(x)\) and \(r\) for the function to be rewritten as \(f(x)=(x- (2+ \sqrt{2})) q(x) + r\). Verifying this equation for \(x=2+\sqrt{2}\), one should also confirm that \(f(2+\sqrt{2})=r\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that provides a straightforward way to find the remainder of a polynomial division. When a polynomial \(f(x)\) is divided by a linear divisor \((x-k)\), the theorem states that the remainder of this division is simply \(f(k)\).
This means that instead of going through the entire polynomial division process to find the remainder, you can directly substitute the value \(k\) into the function and calculate \(f(k)\).
This is not only a huge time saver but also provides a quick verification method to check if a divisor is a factor of the polynomial.

• If \(f(k) = 0\), then \((x-k)\) is a factor of \(f(x)\).
• The Remainder Theorem helps in confirming that \(f(k)\) equals \(r\) when dividing \(f(x) = (x-k)q(x) + r\).

Understanding and applying the Remainder Theorem can greatly simplify solving polynomial equations and analyzing their divisibility.

Algebraic Functions

Algebraic functions are functions that involve polynomials, and possibly roots of polynomials, in their expressions. These functions can range from simple linear functions like \(f(x) = 2x + 3\) to much more complicated expressions involving radicals.
Algebraic functions arise frequently in mathematical problems and applications, providing the backbone for expressing relationships in various fields of science and engineering.

• Linear functions such as \(f(x) = mx + b\) are the simplest form of algebraic functions.
• Polynomial functions, like \(f(x) = 15x^4 + 10x^3 - 6x^2 + 14\), are more complex, yet still fall under algebraic functions.
• Equations involving square roots, cube roots, or other radicals are also considered algebraic if they can be expressed in terms of a polynomial equation.

Prospective solutions to algebraic functions often need polynomial division or factoring techniques, especially when dealing with higher-orders. This includes using the Remainder Theorem or synthetic division for quick analysis and simplification.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents of variables.
They are categorized by their degree, which is determined by the highest exponent of the variable in the expression.

• A polynomial of degree zero is a constant (e.g., \(f(x) = 5\)).
• A polynomial of degree one is linear (e.g., \(f(x) = 7x + 2\)).
• Quadratic polynomials have a degree of two (e.g., \(f(x) = 3x^2 - x + 1\)).
• Cubic polynomials, like \(f(x) = x^3 - 6x + 4\), have a degree of three.

Polynomials play a crucial role in areas such as calculus and complex analysis, where they are used to approximate more complicated functions. Mathematical operations on polynomials are simplified by their algebraic structure, facilitating tasks like evaluating algebraic functions or finding roots.

Graphing Utilities

Graphing utilities are tools, either software or physical devices, designed to represent algebraic functions and equations visually on a graph.
These utilities allow students and mathematicians to gain a better understanding of the behavior of algebraic expressions by viewing their graphical representations.

• They help in visualizing solutions of functions—seeing all the real roots, intersections, and asymptotic behaviors.
• Graphing calculators can solve equations, evaluate functions, and provide quick checks on the accuracy of results derived algebraically.
• Software programs like Desmos or GeoGebra offer advanced features such as sliders for real-time manipulation of variables.

Using a graphing utility, one can confirm the results of polynomial division by plotting the original polynomial function alongside its linear divisor and observing their interaction graphically. This visualization can also corroborate findings from the Remainder Theorem and other algebraic tactics.