Problem 75 - College Algebra

Question

When \(2 x^{2}-7 x+9\) is divided by a polynomial, the quotient is \(2 x-3\) and the remainder is \(3 .\) Find the polynomial.

Step-by-Step Solution

Answer

The polynomial is \(D = \frac{2x^2 - 7x + 6}{2x - 3}\).

Step 1: Understand the relationship

In general form, polynomial division can be written as \(dividend = divisor \times quotient + remainder\). In our problem, the dividend is \(2x^2 - 7x + 9\), the quotient is \(2x - 3\), and the remainder is \(3\). We don't know the divisor, so let's call it \(D\). This gives us the equation \(2x^2 - 7x + 9 = D \times (2x - 3) + 3\).

Step 2: Rearrange the equation

To find \(D\), we need to rearrange the equation above. Subtract 3 from both sides to isolate \(D\): \(2x^2 - 7x + 9 - 3 = D \times (2x - 3)\). Simplify the left side: \(2x^2 - 7x + 6 = D \times (2x - 3)\). Finally, divide both sides by \(2x - 3\) to solve for \(D\).

Step 3: Solve for Divisor, D

Now, dividing both sides by \(2x - 3\), we get: \(D = \frac{2x^2 - 7x + 6}{2x - 3}\). This is our polynomial divisor.