Perform the division below.

Question

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 $y$:

   $$y^2÷y=y$$

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 $y$:

   $$-4y÷y=-4$$

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:

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