Use synthetic division to divide.
\[
\frac{7 x^{3}-63 x^{2}+62 x-48}{x-8}
\]
Final Answer: The quotient is \(\boxed{7x^{2}-7x+6}\) and the remainder is \(\boxed{0}\).
Step 1 :Given the polynomial \(7x^{3}-63x^{2}+62x-48\) and the divisor \(x-8\), we are to perform synthetic division.
Step 2 :First, write down the coefficients of the dividend polynomial, which are [7, -63, 62, -48].
Step 3 :Next, write down the value of 'a' from the divisor polynomial \(x - a\), which is 8 in this case.
Step 4 :Bring down the first coefficient to the bottom row, which is 7.
Step 5 :Multiply 'a' by the value just written on the bottom row, and write the result under the next coefficient. Repeat this step until all coefficients have been processed.
Step 6 :The bottom row of numbers represents the coefficients of the quotient polynomial, and the last number is the remainder.
Step 7 :After performing synthetic division, the coefficients of the quotient polynomial are 7, -7, and 6, and the remainder is 0.
Step 8 :This means that the original polynomial is exactly divisible by \(x - 8\), and the quotient is \(7x^{2} - 7x + 6\).
Step 9 :Final Answer: The quotient is \(\boxed{7x^{2}-7x+6}\) and the remainder is \(\boxed{0}\).