Problem 58 - 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. $$\frac{2 x^{3}-3 x^{2}-3 x+4}{x-1}=2 x^{2}-x+4, x \neq 1$$

Step-by-Step Solution

Answer

The actual procedure in the exercise is to confirm if a polynomial division has been performed correctly by graphing both sides of the equation. If the graphs do not coincide, the polynomial needs to be corrected. The exercise provides good practice for polynomial long division and using a graphing utility to verify the result.

Step 1: Graphing the function on both sides

By using a graphing utility, set two equations \(y= \frac{2x^{3}-3x^{2}-3x+4}{x-1}\) and \(y= 2x^{2}-x+4\) to observe if the graphs coincide.

Step 2: Evaluating the expressions

If it is observed that the graphs do not overlap, we can conclude that the division hasn't been performed correctly.

Step 3: Performing long polynomial division to correct the right expression

Perform long division of the polynomial \(2x^{3}-3x^{2}-3x+4\) by \(x-1\), to get the correct expression. The result of this division will be a polynomial of degree 2.

Step 4: Graphing the correct function

After correction, draw the graph of both functions again using the same graphing utility. The graphs of both functions should now coincide.

Key Concepts

Graphing Utility

Utilizing a graphing utility can help us visualize functions easily. When testing if a division is correct, graph both resultant functions from the equation. For example, if you want to check whether your polynomial division of \(\frac{2x^{3}-3x^{2}-3x+4}{x-1}\) resulted in \(2x^{2}-x+4\), you graph both.

• Set the first function, \(y = \frac{2x^3 - 3x^2 - 3x + 4}{x-1}\).
• Set the second function, \(y = 2x^2 - x + 4\).

Observe both graphs within the same window. If they overlap perfectly, your division is likely correct. If not, some correction is needed.

Polynomial Long Division

Polynomial long division is similar to numerical long division. You divide a polynomial by another smaller degree polynomial, systematically performing subtraction and distributing tokens.To apply this to \(2x^3 - 3x^2 - 3x + 4\) divided by \(x-1\):1. Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\) to get \(2x^2\).2. Multiply the entire divisor by \(2x^2\), getting \(2x^3 - 2x^2\) and subtract from the original polynomial.3. Bring down the next term and repeat until you reach the constant term. Each iteration yields a term in the quotient until a remainder or zero remains.

Equation Correction

Once you detect errors in your polynomial division through graphing, perform polynomial long division to correct them. Suppose the expected quotient doesn't coincide with your polynomial. The graph will showcase discrepancies, indicating an error.

• Use your calculated quotient from division.
• If division reveals a remainder, verify that every term was accounted for accurately.

The correct equation ensures both functions fully coincide when plotted after corrections.

Graph Comparison

Graph comparison implies examining overlays of graphical depictions of two mathematical functions. Start by graphing your original equation and your result-side equation. Align both in one graph space to visually inspect accuracy. If both graphs are mirror images in terms of their plot:

• The division correctness is confirmed visually.
• For a non-coinciding graph, utilize long division correction steps.

Exact comparison helps ensure not only theoretical correctness but practical visibility as well. Correcting ensures graphs fit into the same pattern as expected.