Question

1. $(p^3 - 17p - 13) div (p + 4)$

Explanation:

Step1: Use polynomial long division. Divide the leading term of the dividend \( p^3 - 17p - 13 \) (which is \( p^3 \)) by the leading term of the divisor \( p + 4 \) (which is \( p \)) to get the first term of the quotient, \( p^2 \).

Multiply the divisor \( p + 4 \) by \( p^2 \): \( p^2(p + 4)=p^3 + 4p^2 \)
Subtract this from the dividend: \( (p^3 - 17p - 13)-(p^3 + 4p^2)= - 4p^2 - 17p - 13 \)

Step2: Divide the leading term of the new dividend \( -4p^2 \) by the leading term of the divisor \( p \) to get the next term of the quotient, \( -4p \).

Multiply the divisor \( p + 4 \) by \( -4p \): \( -4p(p + 4)=-4p^2 - 16p \)
Subtract this from the new dividend: \( (-4p^2 - 17p - 13)-(-4p^2 - 16p)= - p - 13 \)

Step3: Divide the leading term of the new dividend \( -p \) by the leading term of the divisor \( p \) to get the next term of the quotient, \( -1 \).

Multiply the divisor \( p + 4 \) by \( -1 \): \( -1(p + 4)=-p - 4 \)
Subtract this from the new dividend: \( (-p - 13)-(-p - 4)= - 9 \)
So, the result of the division \( (p^3 - 17p - 13)\div(p + 4) \) is the quotient \( p^2 - 4p - 1 \) with a remainder of \( -9 \), which can be written as \( p^2 - 4p - 1-\frac{9}{p + 4} \)

Answer:

\( p^2 - 4p - 1-\frac{9}{p + 4} \)