Problem 59 - 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. $$\frac{3 x^{4}+4 x^{3}-32 x^{2}-5 x-20}{x+4}=3 x^{3}+8 x^{2}-5, x \neq-4$$

Step-by-Step Solution

Answer

The method is by graphing both sides, and if they do not coincide, perform polynomial long division. Then verify again with the graphing tool.

Step 1: Graph the Given Functions

Use a graphing utility to graph both the left side \(\frac{3 x^{4}+4 x^{3}-32 x^{2}-5 x-20}{x+4}\) and the right side of the equation \(3 x^{3}+8 x^{2}-5\). If these two functions produce the same graph, then the division is correct.

Step 2: Long Division

If the graph doesn't coincide, it's essential to perform polynomial long division to verify the correctness of the right side of the equation. Arrange the terms in order of degree from highest to lowest and divide the polynomial \(3 x^{4}+4 x^{3}-32 x^{2}-5 x-20\) by \(x+4\). The result should be the correct expression.

Step 3: Verification

Substitute the new expression obtained from the long division into the equation, and then use the graphing utility to verify the solution. If the graphs of left side (new expression) and right side of the equation coincide, then division has been performed correctly.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to how you would divide numbers. It helps simplify the dividend by breaking it down using the divisor, resulting in a quotient and sometimes a remainder. To perform polynomial long division, follow these steps:

• Arrange both the dividend and divisor in descending order of their terms.
• Divide the leading term of the dividend by the leading term of the divisor. This quotient is the first term of your result.
• Multiply the entire divisor by this first term of the quotient and subtract the result from your dividend.
• Bring down the next term from the original dividend to the result of the subtraction and repeat the process until all terms have been brought down.
• If there is a non-zero remainder, it should be expressed as a fraction along with the quotient.

With polynomial long division, you can check the accuracy of the algebraic expression obtained from a division, especially when you need to determine if it matches another given polynomial.

Function Graphing

Function graphing is essential for visualizing polynomial functions and understanding their behavior over different values of the variable. By graphing functions, you can easily compare outputs and verify equations. Here's how to effectively graph functions:

• Choose your graphing utility, like a graphing calculator or software.
• Input the polynomial function equations. In this case, input \(\frac{3 x^{4}+4 x^{3}-32 x^{2}-5 x-20}{x+4}\) and \(3 x^{3}+8 x^{2}-5\).
• Set the same viewing window for both graphs to ensure accurate comparison.
• Analyze the graphs. If both functions coincide (they appear as one graph), the polynomial division is correct. If not, a correction using polynomial long division is necessary.

Graphing is not just about confirming calculations; it's also a powerful tool for identifying errors in polynomial relationships.

Polynomial Division Verification

Polynomial division verification is the process of confirming that your polynomial division results are accurate using graphical or algebraic methods. This verification is crucial in complex polynomial operations to avoid errors. Here’s how you can effectively verify:

• Use the result from polynomial long division as the new expression.
• Input this new expression into your graphing utility as one of the functions.
• If the graphs of the newly calculated expression and the given expression match, this indicates that the division was completed correctly.

This method saves time by preventing the need for repeated algebraic manipulation and provides a visual confirmation of correctness. The graphical approach of verification serves as a reassuring checkpoint for ensuring the integrity of polynomial calculations.