Problem 57 - College Algebra

Question

Explain how to perform long division of polynomials. Use \(2 x^{3}-3 x^{2}-11 x+7\) divided by \(x-3\) in your explanation.

Step-by-Step Solution

Answer

The result of the division of \(2 x^{3}-3 x^{2}-11x+7\) by \(x-3\) is \(2x^2 + 3x - 8 + \frac{1}{x-3}\).

Step 1 Initiate the Division

The division process of \(2 x^{3}-3 x^{2}-11x+7\) by \(x-3\) is commenced by dividing the first term (leading term) of the numerator by the first term of the denominator, which gives \(2x^2\).

Step 2 Multiply and Subtract

Next, \(x-3\) is multiplied by the result obtained in step 1, \(2x^2\), yielding \(2x^{3} - 6x^{2}\). Then, subtract this from the original numerator \(2 x^{3}-3 x^{2}-11x+7\), which gives the new numerator \(3x^2 - 11x + 7\).

Step 3 Repeat the Process

Now, the process should be repeated by dividing the leading term of the updated numerator by the leading term of the denominator, yielding the next term of the quotient, 3x. Then, the denominator \(x-3\) should be multiplied by the newly obtained term 3x, and the result is subtracted from the updated numerator. This results in the new numerator \( - 8x + 7\).

Step 4 Finish the Long Division

Repeat the same operation: divide -8x by x to obtain -8, and multiply \(x-3\) by -8. Then subtract this from the final numerator to get a remainder of 1. Thus, the result of the division, or the quotient is \(2x^2 + 3x -8\) with the remainder 1.

Step 5 Append the Remainder

Include the remainder as a fraction over the divisor to the end of the quotient, so the full representation of the solution becomes \(2x^2 + 3x - 8 + \frac{1}{x-3}\).