Problem 119 - Introductory Algebra for College Students

Question

Use a graphing utility to graph each side of the equation in the same viewing rectangle. Do the graphs coincide? If so, this means that the polynomial on the left side has been factored correctly. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{2}-2 x+5 x-10=(x-2)(x-5)$$

Step-by-Step Solution

Answer

The provided equation was not factored correctly. The right factorization is \( (x-2)(x+5) \). After correctly factoring the equation and graphing, both graphs match, verifying the correct factorization.

Step 1: Left Hand Side Polynomial Graph

First, plug in values for \( x \) into the left-hand side of the equation \( x^{2}-2 x+5 x-10 \) to achieve a range of values of \( y \) (or the output of the function). Input these data points into a graphing utility and plot the function.

Step 2: Right Hand Side Polynomial Graph

Next, plug in the same values for \( x \) into the right-hand side of the equation \( (x-2)(x-5) \). The output will give the corresponding \( y \) values. Use a graphing utility to plot these data points on the same graph as the left-hand side function.

Step 3: Compare the Graphs

Now compare the two graphs. If they coincide i.e., if they are identical, then the equation has been factored correctly and the task ends here. If not, proceed to the next step for factoring correction.

Step 4: Correct Factorization

If the graphs don't match, it means that the polynomial has not been factored correctly. In this case, one needs to factor the left-hand side equation properly. The correct factored form of \( x^{2}-2 x+5 x-10 \) is \( (x-2)(x+5) \) after rearranging as \( x^{2}+3 x-10 \).

Step 5: Verify Correct Factorization

Finally, graph the function \( (x-2)(x+5) \). If the new graph coincides with the initial one plotted for the left-hand side, it verifies that the factoring correction was executed correctly.