Question

Perform the division below.
StartFraction y squared plus 2 y plus 11 Over y plus 6 EndFraction

Answer 1

To perform the division of the polynomial \( \frac{y^2 + 2y + 11}{y + 6} \), we can use polynomial long division.

1. Divide the leading term of the numerator \( y^2 \) by the leading term of the denominator \( y \):

\[
y^2 \div y = y
\]

2. Multiply the entire divisor \( y + 6 \) by this result \( y \):

\[
y(y + 6) = y^2 + 6y
\]

3. Subtract this from the original numerator:

\[
(y^2 + 2y + 11) - (y^2 + 6y) = 2y + 11 - 6y = -4y + 11
\]

4. Now, we take the new polynomial \(-4y + 11\) and repeat the process. Divide the leading term \(-4y\) by \(y\):

\[
-4y \div y = -4
\]

5. Multiply the entire divisor \(y + 6\) by \(-4\):

\[
-4(y + 6) = -4y - 24
\]

6. Subtract this from \(-4y + 11\):

\[
(-4y + 11) - (-4y - 24) = 11 + 24 = 35
\]

Now, we are left with a remainder of \(35\).

Combining all parts, the result of the division is:

\[
y - 4 + \frac{35}{y + 6}
\]

So, the final answer is:

\[
\frac{y^2 + 2y + 11}{y + 6} = y - 4 + \frac{35}{y + 6}
\]