Problem 41 - 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 $$f(x)=4 x^{3}-6 x^{2}-12 x-4$$

Step-by-Step Solution

Answer

The given exercise is a long process when done for all values of 'k'. Nevertheless, the solution basically requires rewriting the polynomial in the form \(f(x)=(x-k) q(x)+r(x)\), performing long division, and validating the result with a graphing utility.

Step 1: Write Down the Polynomial and the Value of k

Given the polynomial \(f(x)=4 x^{3}-6 x^{2}-12 x-4\) and we have multiple values of \(k\) to test, for example, when \(k=4\), we begin by substituting \(k\) into the function's equation, forming \(f(k)=(k-4) q(k)+r\).

Step 2: Perform Long Division

Perform polynomial long division, treating \((x-k)\) as the divisor and \(f(x)\) as the dividend. For \(k=4\), it would be \((x-4)\) to be used as divisor. Resulting quotient and remainder would form \(q(x)\) and \(r(x)\) respectively.

Step 3: Validate with a Graphing Utility

Examine the correctness of this result by using a graphing utility to display \(f(k)\) and \(r\), wherein, if these two are equal, then the prior steps have been executed correctly.

Step 4: Repeat for All Values of k

Repeat the process for all remaining k values provided in the exercise. \(k=-2/3\), \(k= sqrt(2)\), \(k=-sqrt(5)\), \(k=1-sqrt(3)\), and \(k=2+sqrt(2)\)

Key Concepts

Polynomial Function

Polynomial functions are one of the most commonly encountered types of functions in algebra and calculus. They are formed by combining terms consisting of a variable raised to a non-negative integer exponent with real number coefficients. Such a function can be expressed in the form
\(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(a_n\) are coefficients and \(n\) is a non-negative integer. The highest exponent in the function denotes the degree of the polynomial.
When faced with a polynomial like \(f(x)=4x^3-6x^2-12x-4\) from this exercise, it's essential to understand how these terms contribute to the shape and behavior of its graph, among other properties such as end behavior, intercepts, and turning points. A deep understanding of polynomial functions is crucial for solving algebraic problems as well as grasping more advanced concepts in calculus, such as finding local maxima and minima.

Remainder Theorem

The Remainder Theorem is a pivotal concept in algebra, particularly when we are dealing with polynomial division. This theorem states that when a polynomial \(f(x)\) is divided by a linear divisor of the form \((x - k)\), the remainder of this division is equal to the value of \(f(k)\). So rather than performing long division, you could simply substitute \(k\) into the polynomial function to find the remainder.
In our example, dividing \(f(x)=4x^3-6x^2-12x-4\) by \((x-4)\), according to the Remainder Theorem, the remainder should be \(f(4)\), which can be much quicker to compute. Employing this theorem is a neat shortcut that saves both time and effort, but remember – it only works for linear divisors of the form \((x - k)\). Moreover, it has an important role in other concepts like Factor Theorem, which can be used for factorizing polynomials further.

Graphing Utility

Graphing utilities are invaluable tools for visualizing the behavior of mathematical functions, particularly polynomial functions. These utilities take an algebraic equation and transform it into a graphical representation on a coordinate plane. When dealing with an exercise like ours, a graphing utility can be exceptionally beneficial to visually verify the result of a polynomial division.
By entering the function \(f(x)=4x^3-6x^2-12x-4\) and the divisor \((x - k)\) into a graphing utility, one can clearly see the point on the graph where the function intersects the x-axis at \(x=k\). This intersection represents the remainder (\(r\)) when evaluating \(f(k)\) — effectively the same result one would get from the polynomial long division process. A graphing utility offers a powerful visual confirmation, ensuring that the remainder you've calculated algebraically aligns with the graphic depiction of the function.