To determine if [tex]\( p - q \)[/tex] is a factor of the polynomial [tex]\( f(p) = p^2 + (q+3) p^2 - p (3q+3) + 3q \)[/tex], we can perform polynomial division and examine the remainder. If the remainder is zero, then [tex]\( p - q \)[/tex] is a factor of [tex]\( f(p) \)[/tex]; otherwise, it is not. Let's go through the steps of polynomial division:
1. Write down the polynomial to be divided:
[tex]\[
f(p) = p^2 + (q+3)p^2 - p(3q+3) + 3q
\][/tex]
2. Simplify the polynomial:
Combine like terms:
[tex]\[
f(p) = p^2 + (q + 3)p^2 - 3p(q + 1) + 3q \\
f(p) = p^2 + q p^2 + 3p^2 - 3pq - 3p + 3q
\][/tex]
3. Arrange the terms in standard form:
[tex]\[
f(p) = (1 + q + 3)p^2 - 3(pq + p) + 3q
\][/tex]
Simplify it further if possible:
[tex]\[
f(p) = (1 + q + 3)p^2 - 3pq - 3p + 3q
\][/tex]
4. Perform the polynomial division:
Divide [tex]\( f(p) \)[/tex] by [tex]\( p - q \)[/tex].
[tex]\(p - q\)[/tex] into [tex]\((4+q)p^2\)[/tex] goes [tex]\( (q+4)p \)[/tex] times.
Multiply [tex]\( (p-q) \)[/tex] by [tex]\( (q+4)p \)[/tex]:
[tex]\[
(q+4)p(p - q) = (q+4)p^2 - (q+4)pq
\][/tex]
5. Subtract the result from [tex]\( f(p) \)[/tex]:
[tex]\[
\left[(4+q)p^2 - 3pq - 3p + 3q \right] - \left[(4+q)p^2 - (4+q)pq\right]
\][/tex]
Simplify this subtraction:
[tex]\[
(4+q)p^2 - 3pq - 3p + 3q - (4+q)p^2 + (4+q)pq
\][/tex]
[tex]\[
-3pq + (4+q)pq - 3p + 3q
\][/tex]
[tex]\[
(p q + q ^2 - 3)p + 3 q
\][/tex]
6. Perform polynomial division to find the final remainder:
Divide [tex]\( (p q + q^2 - 3)p + 3 q \)[/tex] by [tex]\( p-q \)[/tex].
It goes once as [tex]\( (p q + q ^2) \)[/tex], subtracting this gives zero remainder.
Thus, the quotient is [tex]\( p q + 4 p + q^2 + q - 3 \)[/tex], and the remainder is [tex]\( q^3 + q^2 \)[/tex].
Since the remainder is not zero, this completes our proof that [tex]\( p - q \)[/tex] is not a factor of [tex]\( f(p) \)[/tex].
[tex]\[
\boxed{q^3 + q^2}
\][/tex]
Hence, [tex]\( p - q \)[/tex] is not a factor of the given polynomial [tex]\( f(p) = p^2 + (q+3)p^2 - p(3q+3) + 3q \)[/tex].
