Problem 115 - A Graphical Approach to Precalculus with Limits

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

As |x| → ∞, f(x) approaches g(x) because the remainder becomes negligible.

Step 1: Set Up the Division

Write the polynomial division problem as \\[x^{4} - 5x^{2} + 4 \div (x^{2} + x - 12)\]. \Represent it in a division bracket where \(x^{4} - 5x^{2} + 0x + 4\) is inside the bracket, and \(x^{2} + x - 12\) is outside.

Step 2: Perform the First Division

Divide the first term of the numerator \(x^{4}\) by the first term of the denominator \(x^{2}\) to get \(x^{2}\). Multiply \(x^{2}\) by \(x^{2} + x - 12\), giving \(x^{4} + x^{3} - 12x^{2}\). Subtract this from \(x^{4} - 5x^{2}\) to get a new polynomial: \(-x^{3} + 7x^{2} + 0x + 4\).

Step 3: Perform the Second Division

Divide the new first term \(-x^{3}\) by \(x^{2}\) to get \(-x\). Multiply \(-x\) by \(x^{2} + x - 12\) to get \(-x^{3} - x^{2} + 12x\). Subtract this from \(-x^{3} + 7x^{2} + 4\), resulting in \(8x^{2} - 12x + 4\).

Step 4: Perform the Third Division

Divide \(8x^{2}\) by \(x^{2}\) resulting in \(8\). Multiply \(8\) by \(x^{2} + x - 12\) to get \(8x^{2} + 8x - 96\). Subtract this from \(8x^{2} - 12x + 4\) to obtain the remainder, which is \(-20x + 100\).

Step 5: Verification of Quotient and Remainder

The quotient of the division is \(x^{2} - x + 8\) and the remainder is \(-20x + 100\), confirming the given solution.

Step 6: Graph the Functions

Graph \(f(x) = \frac{x^{4}-5x^{2}+4}{x^{2}+x-12}\) and \(g(x) = x^{2} - x + 8\) in the window \([-50, 50]\) by \([0, 1000]\). Note that \(g(x)\) represents the asymptotic behavior of \(f(x)\), with the two graphs appearing closer together as \(|x|\) becomes large.

Step 7: Analyze the Graphs

Observe that as \(|x| \to \infty\), the difference between \(f(x)\) and \(g(x)\) is determined by the remainder term \(\frac{-20x + 100}{x^{2} + x - 12}\), which approaches zero. This indicates that \(f(x)\) and \(g(x)\) become nearly indistinguishable for large \(|x|\). Thus, \(f(x)\) approaches \(g(x)\) as \(|x| \to \infty\).

Key Concepts

Asymptotic Behavior

The asymptotic behavior of a function refers to how it behaves as the input values become very large or very small, often approached at infinity. This is crucial for understanding complex functions like rational functions, which involve polynomial division.
For the function \( f(x) = \frac{x^4 - 5x^2 + 4}{x^2 + x - 12} \), we apply long division to obtain a quotient \( g(x) = x^2 - x + 8 \) and a remainder \(-20x + 100\). As \(|x| \to \infty\), the influence of the remainder diminishes because it is divided by a rapidly increasing polynomial in the denominator. Thus,

• The graph of \(f(x)\) will resemble \(g(x) = x^2 - x + 8\) for very large or very small \(x\).
• This makes \(g(x)\) the horizontal asymptote of \(f(x)\).

In essence, despite \(f(x)\) and \(g(x)\) having different forms in general, at extreme values of \(x\), their graphs become nearly indistinguishable.

Graphical Representation

Graphing functions helps visualize how they behave, particularly over a specified interval. By graphing both \(f(x)\) and \(g(x)\), we can see their relationship, including how closely the two functions approach each other.
The goal here is to observe how \(f(x)\) behaves compared to its quotient \(g(x)\) as \(x\) grows larger in magnitude. Specifically:

• By plotting \(f(x)\) and \(g(x)\) in the window \([-50, 50]\) along the \(x\)-axis, and \([0, 1000]\) along the \(y\)-axis, you'll notice how closely \(f(x)\) follows \(g(x)\) as \(x\) moves toward these extremes.
• The smaller the remainder compared to the quotient, the closer \(f(x)\) and \(g(x)\) will appear on the graph.

Visualizing the graphs of these functions provides an intuitive sense of their asymptotic relationship.

Polynomial Division

In mathematics, polynomial division is akin to numerical long division, helping us simplify complex expressions. When dividing polynomials, like in the original exercise, our goal is to separate the expression into a simpler quotient and a remainder.
This process involves several key steps:

• Identify the divisor and the dividend, ensuring all polynomial terms are present (even those with zero coefficients).
• Divide the leading term of the dividend by the leading term of the divisor to find each term of the quotient.
• Subtract the resulting product from the dividend, bringing down subsequent terms until reaching the end. The final leftover terms form the remainder.

Through this process, we learn that:

• The quotient \(x^2 - x + 8\) describes how many times the divisor \((x^2 + x - 12)\) fits into the dividend.
• The remainder \(-20x + 100\) arises from what does not fit perfectly, helping understand how close \(f(x)\) is to simply being a polynomial.

Remainder Theorem

The Remainder Theorem plays a critical role in polynomial division. It tells us that if a polynomial \(f(x)\) is divided by \(x - c\), then the remainder of that division is simply \(f(c)\).
In polynomial long division where the divisor is of higher degree, like in our example \(x^2 + x - 12\), the remainder becomes a function of lesser degree than the divisor. This remainder tells us how 'far off' the division result is from an exact quotient.
For this problem:

• The remainder \(-20x + 100\) indicates that after dividing \(f(x)\) by \(x^2 + x - 12\), there is a part left over.
• This part gets smaller as \(x\) becomes very large or very negative, leading to \(f(x)\) closely approximating \(g(x) = x^2 - x + 8\) for such values.

The observations help predict the end behavior of graphed polynomial functions.