Questions: Consider the expression (2y^2-8y-15)/(y-5). Divide, using the polynomial long division algorithm. Fill in your work below.

Solution Steps

To divide the polynomial \( \frac{2y^2 - 8y - 15}{y - 5} \) using polynomial long division, we will follow these steps:

1. Divide the leading term of the dividend \(2y^2\) by the leading term of the divisor \(y\) to get the first term of the quotient.
2. Multiply the entire divisor by this term and subtract the result from the original dividend.
3. Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.

We start with the polynomial expression given by the dividend \(2y^2 - 8y - 15\) and the divisor \(y - 5\).

To divide \(2y^2 - 8y - 15\) by \(y - 5\), we first divide the leading term of the dividend \(2y^2\) by the leading term of the divisor \(y\): \[ \frac{2y^2}{y} = 2y \] This gives us the first term of the quotient.

Next, we multiply the entire divisor \(y - 5\) by \(2y\): \[ 2y(y - 5) = 2y^2 - 10y \] Now, we subtract this result from the original dividend: \[ (2y^2 - 8y - 15) - (2y^2 - 10y) = 2y - 15 \]

Now we take the new polynomial \(2y - 15\) and divide it by the divisor \(y - 5\). We divide the leading term \(2y\) by \(y\): \[ \frac{2y}{y} = 2 \] This gives us the next term of the quotient.

We multiply the divisor \(y - 5\) by \(2\): \[ 2(y - 5) = 2y - 10 \] Subtracting this from \(2y - 15\) gives: \[ (2y - 15) - (2y - 10) = -5 \]

Final Answer