Question

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=2 x^{3}+x^{2}-14 x-10, \quad k=1+\sqrt{3}$$

Short answer

After performing polynomial division, you should find \(q(x)\) and the remainder \(r\). The function will be in the form \(f(x) = (x - k) q(x) + r\). Finally, by substituting the given value of \(k\), you'll check that \(f(k)\) equals to the remainder \(r\), effectively demonstrating the required condition.

Step-by-step solution

Polynomial Division

Perform the polynomial division of \(f(x)\) by \((x - k)\) which is \((x - (1 + \sqrt{3}))\). The outcome of this step will be a quotient \(q(x)\) and a remainder \(r\). This will arrange the function in form \(f(x) = (x - k) q(x) + r\). You can perform this division similarly as you would perform long division for integers.

Expand \((x - k) q(x)\)

Expand the product \((x - k) q(x)\) obtained in the previous step. Collect like terms and compare the coefficients on the right and left side. Your remainder \(r\) is the constant term (the term which does not involve \(x\)).

Validate if \(f(k) = r\)

Substitute \(k = 1 + \sqrt{3}\) into the original function \(f(x)\). The aim here is to check if the function evaluates to \(r\), i.e., if \(f(k) = r\). If this equation holds true, it verifies that the function has been correctly written in the form \(f(x) = (x - k) q(x) + r\).

Key concepts

Remainder Theorem

The Remainder Theorem is a fundamental aspect of algebra that provides a quick way for evaluating the remainder of a polynomial divided by a linear divisor of the form \(x-k\). According to the theorem, if a polynomial \(f(x)\) is divided by \(x-k\), the remainder is simply \(f(k)\). This principle allows you to verify the results of polynomial division without conducting the entire division process.

For example, given a polynomial \(f(x)=2x^3+x^2-14x-10\), and we wish to know the remainder when it is divided by \(x-(1+\sqrt{3})\), we simply evaluate \(f(1+\sqrt{3})\). This evaluation gives us the remainder directly, thus confirming the result of the polynomial division. The Remainder Theorem is incredibly useful for checking the work done by polynomial long division and synthetic division methods.

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial, typically of a lower degree. The process is analogous to the long division of integers, arranged in a similar tabular format. The main goal is to determine the quotient \(q(x)\) and the remainder \(r\) when dividing a polynomial \(f(x)\) by a divisor, such as \(x-k\).

The steps involve dividing the leading term of \(f(x)\) by the leading term of the divisor, multiplying the entire divisor by this result, subtracting it from \(f(x)\), and repeating this process with the new polynomial formed after the subtraction. The division concludes when the degree of the remaining polynomial is less than the degree of the divisor. This remaining term is the remainder \(r\).

The key to understanding polynomial long division is practice. It's important to follow each step carefully, keep your work organized, and consistently check that each subtraction and multiplication is completed correctly. Clear presentation of work and attention to detail are critical for success with this technique.

Synthetic Division

Synthetic division is a simplified form of polynomial division that's particularly useful when dividing by a linear factor. It's a shortcut that reduces the amount of written work. Instead of using the long division layout, you write only the coefficients of the polynomial and perform the division algorithm in a compact computation.

Let's consider dividing a polynomial like \(2x^3 + x^2 - 14x - 10\) by a factor like \(x - k\), where \(k\) is a constant. The steps of synthetic division involve bringing down the leading coefficient, multiplying it by \(k\), and adding it to the next coefficient. This process is repeated until you reach the last coefficient. The final number obtained is the remainder of the division.

It's important to note that synthetic division can only be used when the divisor is of the first degree and when the leading coefficient is 1. For any higher-degree or more complex divisors, one must revert to polynomial long division.

Function Transformation

Function transformation refers to the various ways a function can be altered or 'transformed' to produce a new function. This includes shifting, stretching, or reflecting the graph of the original function. When we express a polynomial as \(f(x) = (x - k) q(x) + r\), we are essentially transforming it into a different format that can provide insights into the behavior of the polynomial, such as its roots and end behavior.

By factoring out \(x - k\), we emphasize the transformation that shifts the original function horizontally by \(k\). When \(k\) is a root of the polynomial, this transformation reflects that the graph of the function will touch or cross the x-axis at \(x=k\). The remaining part of the transformed function, embodied in the quotient \(q(x)\), and the remainder \(r\), can exhibit other interesting features about the polynomial, such as its degree and general shape.

Understanding how to apply and interpret these transformations is essential for grasping more complex concepts in calculus and higher-level mathematics. It allows students to transition from the algebraic manipulation of polynomials to the graphical interpretations that are vital for thorough mathematical analysis.