Problem 57 - College Algebra

Question

Use a graphing utility to determine if the division has been performed correctly Graph the function on each side of the equation in the same viewing rectangle. If the graphs do not coincide, correct the expression on the right side by using polynomial long division. Then verify your correction using the graphing utility. $$\begin{aligned} &\frac{x^{4}+6 x^{3}+6 x^{2}-10 x-3}{x^{2}+2 x-3}=x^{2}+4 x+1, x \neq-3, x \neq 1 \end{aligned}$$

Step-by-Step Solution

Answer

If the graphs of the given functions do not coincide, perform polynomial long division and graph the resulting function for verification. Keep in mind the provided constraints \(x \neq-3\) and \(x \neq 1\).

Step 1: Graph the equations

Start by graphing both \(\frac{x^{4}+6 x^{3}+6 x^{2}-10 x-3}{x^{2}+2 x-3}\) and \(x^{2}+4 x+1\). Observe if the graphs coincide. The provided values claim that \(x \neq-3, x\neq 1\), so keep these constraints in mind

Step 2: Perform Polynomial Long Division

If the graphs do not coincide, apply polynomial long division to the equation. Divide \(x^{4}+6 x^{3}+6 x^{2}-10 x-3\) by \(x^{2}+2 x-3\). The result of this division will be the correct function which must be graphed

Step 3: Graph the Resulting Function

Graph the function obtained from the long division and verify if this coincides with the original graph

Step 4: Verification

Lastly, confirm that the results from long division and the graph are indeed correct. This confirms that the division was performed correctly.

Key Concepts

Graphing Utility

Graphing utilities are essential tools used to visualize complex algebraic expressions, including polynomials and rational functions. By inputting a function into a graphing utility, such as a calculator or software, you can quickly see the shape and behavior of the function's graph. This becomes particularly useful when you need to verify solutions to algebraic problems.

For example, when you're given an equation like the one in the exercise, graphing the left and right sides of the equation separately helps you compare their outputs visually. If the two sides of the equation produce identical graphs, the division has been done correctly. Otherwise, the discrepancy in the graphs prompts the need for re-evaluation using methods like polynomial long division.

Incorporating graphing utilities in confirming algebraic manipulations helps enhance understanding and provides a visual check, which is an invaluable asset for students learning about functions and their properties.

Verifying Algebraic Expressions

After performing polynomial long division, it's crucial to verify that the quotient correctly represents the division of the original polynomial by the divisor. One verification method aside from the graphing utility is substituting arbitrary values for the variable and checking whether the original expression equals the quotient multiplied by the divisor plus any remainder.

Verification can also be approached by revisiting each step of the long division process, ensuring that calculations like subtraction and multiplication of terms have been done accurately. Scrutinizing for common errors like sign mistakes or incorrect term alignment can often reveal where a division might have gone awry. Proper verification confirms the integrity of the algebraic process and solidifies student understanding.

Rational Functions

A rational function is defined as the ratio of two polynomials where the denominator is not equal to zero. In the given exercise, \(\frac{x^{4}+6x^{3}+6x^{2}-10x-3}{x^{2}+2x-3}\) is a rational function with specific excluded values \(x eq -3, x eq 1\) to prevent the denominator from being zero.

Rational functions often depict interesting behaviors near their excluded values, including vertical asymptotes, holes, or horizontal asymptotes. Understanding these components is vital for analyzing the function's graph. Learning how to handle rational functions includes mastering polynomial long division to simplify them and analyzing their graphs, which depict their nuanced behavior.

Function Graph Analysis

Analyzing the graph of a function involves more than just recognizing its shape. It includes understanding its key features: intercepts, asymptotes, end behavior, and any discontinuities such as holes or jumps.

When analyzing graphs, especially those resulting from rational functions, it's important to note the regions where the function is defined or undefined, the behavior as it approaches excluded values, and the overall trend as \(x\) becomes very large or very small. In the context of the exercise, analyzing the graph helps confirm the correctness of the polynomial long division by ensuring the graph of the quotient function aligns with the original function, except at the excluded values.