Problem 97 - A Graphical Approach to College Algebra

Question

Use long division of polynomials to show that for $$f(x)=\frac{x^{4}-5 x^{2}+4}{x^{2}+x-12}$$ if we divide the numerator by the denominator, then the quotient polynomial is \(x^{2}-x+8,\) and the remainder is \(-20 x+100 .\) Graph both \(f(x)\) and \(g(x)=x^{2}-x+8\) in the window \([-50,50]\) by \([0,1000] .\) Comment on the appearance of the two graphs. Explain how the graph of \(f\) approaches that of \(g\) as \(|x| \rightarrow \infty\).

Step-by-Step Solution

Answer

The graphs show that \(f(x)\) approaches \(g(x) = x^2 - x + 8\) as \(|x|\) increases, since the remainder becomes negligible.

Step 1: Set Up the Division

Write the division as a polynomial long division problem, with the divisor \(x^2 + x - 12\) and the dividend \(x^4 - 5x^2 + 4\). Ensure both polynomials are written in standard form with all terms included.

Step 2: Divide the First Terms

Divide the first term of the dividend \(x^4\) by the first term of the divisor \(x^2\) to get \(x^2\). This will be the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(x^2\) by the entire divisor \(x^2 + x - 12\) to get \(x^4 + x^3 - 12x^2\). Subtract this from the current dividend \(x^4 - 5x^2 + 4\), resulting in \(-x^3 + 7x^2 + 4\) as the new dividend.

Step 4: Repeat Division

Divide the new first term of the dividend \(-x^3\) by \(x^2\) to get \(-x\). Add \(-x\) to the quotient and multiply \(-x\) by the divisor to get \(-x^3 - x^2 + 12x\). Subtract to get \(8x^2 - 12x + 4\).

Step 5: Repeat Division Again

Divide \(8x^2\) by \(x^2\) to get \(8\). Add \(8\) to the quotient and multiply \(8\) by the divisor to get \(8x^2 + 8x - 96\). Subtract to find the remainder \(-20x + 100\).

Step 6: State the Result

The division yields a quotient of \(x^2 - x + 8\) and a remainder of \(-20x + 100\). Thus, \(f(x)\) can be expressed as \(x^2 - x + 8 + \frac{-20x + 100}{x^2 + x - 12}\).

Step 7: Graphing the Functions

Graph the quotient function \(g(x) = x^2 - x + 8\) which is a simple parabola. Graph \(f(x)\) that includes the remainder. On the same coordinate window \([-50,50]\) by \([0,1000]\), both graphs should be plotted.

Step 8: Analyze the Graphs

As \(|x|\) becomes very large, the \(\frac{-20x + 100}{x^2 + x - 12}\) part of \(f(x)\) becomes very small, so the graph of \(f(x)\) closely approaches the graph of \(g(x)\), making \(f(x)\) approximate \(g(x)\) for large \(x\).

Key Concepts

Long Division of Polynomials

Long division of polynomials is a process that resembles numerical long division in arithmetic. The goal is to divide one polynomial, the dividend, by another polynomial, the divisor, to get a quotient and a remainder. For the division to be possible, both the dividend and the divisor need to be arranged in descending order of their powers or degrees.

Here’s how you can visualize the process:

• **Setting up the division equation**: Place the dividend polynomial inside the division symbol and the divisor on the outside.
• **Dividing the highest degree terms**: Start by dividing the highest degree term of the dividend by the highest degree term of the divisor. This gives you the first term of the quotient.
• **Multiplying and subtracting**: Multiply the entire divisor polynomial by the term obtained in the previous step. Subtract this from the dividend to get a new polynomial.
• **Repeat the process**: Continue dividing the new highest degree term by the divisor, multiply, and subtract until all terms of the dividend have been processed.

This systematic approach continues until you end up with a remainder that is of a lower degree than the divisor.
The process allows you to express any rational function, such as \( f(x) = \frac{x^{4}-5 x^{2}+4}{x^{2}+x-12} \), as a polynomial quotient plus a remainder over the original divisor: \( f(x) = x^2 - x + 8 + \frac{-20x + 100}{x^2 + x - 12} \). This expression is crucial for understanding how the function behaves over different values of \(x\).

Quotient and Remainder

After completing the long division of polynomials, you'll find two parts: the quotient and the remainder. The quotient polynomial is what you get when the highest degree terms have been divided out, and it can stand alone as a new polynomial. In our exercise, the polynomial division of \( f(x) \) yields a quotient \( g(x) = x^2 - x + 8 \).

The remainder, here \(-20x + 100\), comprises terms that couldn't be fully divided by the divisor and gives insight into how much is "leftover" after division. When the original polynomial is expressed with a remainder, it's written as:

• **Quotient part**: Reflects the overall behavior of the polynomial division.
• **Remainder part**: Shows the leftover when the polynomial is not fully divisible. It's usually expressed over the divisor.

This kind of expression helps with the analysis of polynomial functions, especially as they relate to division and modular arithmetic in algebra. Knowing the quotient and the remainder allows us to understand and predict the polynomial’s behavior, depending on whether \(x\) approaches infinity or zero.

Graphing Polynomial Functions

Graphing polynomial functions is key to visualizing their behavior. It helps to see how functions compare across different values of \(x\), especially at their extrema and asymptotic behavior as \(|x|\) gets very large.

When graphing, consider the following:

• **The shape of the graph**: The degree and the leading coefficient of the polynomial determine the end behavior of the graph.
• **Intercepts**: The points where the graph crosses the x-axis and y-axis are important markers.
• **Local minima and maxima**: These are found by identifying where the function switches direction, which you can find using calculus or graphical methods.

In the exercise, graphing \(f(x)\) and \(g(x) = x^2 - x + 8\) together reveals that, as \(|x|\) becomes large, the effect of the remainder \(\frac{-20x + 100}{x^2 + x - 12}\) diminishes.
This means that \(f(x)\) closely follows the parabolic shape of \(g(x)\), particularly outside of the critical points near the center of the graph.
You can use graphical calculators or software to accurately plot these functions to see how they differ or converge as x moves through different values. Understanding how these graphs relate provides a visual glimpse into the function's behavior, making complex relationships easier to grasp.