A polynomial $P(x)$ and a divisor $d(x)$ are given. Use long division to find the quotient $Q(x)$ and the remainder $R(x)$. Express $P(x)$ in the form $P(x)=d(x) \cdot Q(x)+R(x)$.
\[
\begin{array}{l}
P(x)=x^{3}-343 \\
d(x)=x+7
\end{array}
\]
\[
x^{3}-343=(x+7)(\square)-
\]
Answer
\(\boxed{Q(x) = x^{2} - 7x + 49, R(x) = 0}\)
Step 1 ：We are given a polynomial $P(x) = x^{3} - 343$ and a divisor $d(x) = x + 7$. We are asked to perform polynomial long division to find the quotient $Q(x)$ and the remainder $R(x)$ such that $P(x) = d(x) \cdot Q(x) + R(x)$.
Step 2 ：Performing the polynomial long division, we find that the quotient is $Q(x) = x^{2} - 7x + 49$ and the remainder is $R(x) = 0$.
Step 3 ：Therefore, the polynomial $P(x)$ can be expressed as $P(x) = d(x) \cdot Q(x) + R(x)$, where $Q(x) = x^{2} - 7x + 49$ and $R(x) = 0$.
Step 4 ：\(\boxed{Q(x) = x^{2} - 7x + 49, R(x) = 0}\)
