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Let's analyze the problem step-by-step.

Given:
- When a polynomial \( f(x) \) is divided by \( x - p \), the remainder is \( p^2 \).
- When the same polynomial is divided by \( x - q \), the remainder is \( q^2 \).

We need to find the remainder when \( f(x) \) is divided by \( x^2 - (p+q)x + pq \).

### Solution Approach
Let \( f(x) \) be a polynomial of degree \( n \). If \( f(x) \) is divided by \( x-p \), the remainder theorem tells us that the remainder is \( f(p) \). Given that the remainder is \( p^2 \), we have:

\[
f(p) = p^2.
\]

Similarly, when \( f(x) \) is divided by \( x-q \), the remainder is \( f(q) = q^2 \).

Now, we need to determine the remainder when \( f(x) \) is divided by \( x^2 - (p+q)x + pq \). Notice that:

\[
x^2 - (p+q)x + pq = (x-p)(x-q).
\]

When dividing by \( (x-p)(x-q) \), the remainder will be a linear polynomial, since \( f(x) \) is of degree \( n \geq 2 \). Let the remainder be denoted as \( ax + b \).

This means:

\[
f(x) = (x-p)(x-q) \cdot Q(x) + ax + b,
\]

where \( Q(x) \) is the quotient. To find \( a \) and \( b \), we need to satisfy the conditions:

1. \( f(p) = ap + b = p^2 \),
2. \( f(q) = aq + b = q^2 \).

We now solve these two linear equations simultaneously:

1. \( ap + b = p^2 \),
2. \( aq + b = q^2 \).

Subtracting the first equation from the second:

\[
aq - ap = q^2 - p^2,
\]
\[
a(q-p) = (q-p)(q+p).
\]

Since \( q \neq p \), we can divide both sides by \( q-p \):

\[
a = q+p.
\]

Substitute \( a = q+p \) back into one of the equations, say \( ap + b = p^2 \):

\[
(q+p)p + b = p^2,
\]
\[
qp + p^2 + b = p^2,
\]
\[
b = -qp.
\]

### Final Remainder
The remainder when \( f(x) \) is divided by \( x^2 - (p+q)x + pq \) is:

\[
(q+p)x - pq.
\]

Let me know if you'd like further clarifications or details, and here are some related questions to explore:

1. What would be the remainder if the polynomial was divided by \( x^2 + bx + c \)?
2. How can the Remainder Theorem be used to find the remainder for higher-degree divisors?
3. What is the significance of the polynomial factorization \( (x-p)(x-q) \)?
4. How does the concept of polynomial division change when the divisor has higher degrees?
5. What are the properties of remainders when dividing polynomials of different degrees?

**Tip:** When dividing a polynomial by a divisor of degree \( n \), the remainder will always be of degree less than \( n \).

When a polynomial is divided by x - p, the remainder is p^2. When the same polynomial is divided by x - q, the remainder is q^2. What is the remainder when the polynomial is divided by x^2 - (p + q)x + pq? (p ≠ q).

Find the remainder when a polynomial is divided by x^2 - (p + q)x + pq, given specific remainders for divisions by x - p and x - q. This problem involves the Remainder Theorem and solving simultaneous equations. Ideal for students in Grades 10-12.

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