Questions: Divide. (x^3-6x^2-49)/(x-7)

Solution Steps

To divide the polynomial \(\frac{x^{3}-6 x^{2}-49}{x-7}\), we can use polynomial long division or synthetic division. Here, we'll use synthetic division for simplicity. The steps involve setting up the synthetic division table, performing the division, and obtaining the quotient and remainder.

We start with the polynomial \(x^3 - 6x^2 + 0x - 49\) and the divisor \(x - 7\). The root of the divisor is \(7\).

Using synthetic division, we divide the polynomial by \(x - 7\). The coefficients of the polynomial are \([1, -6, 0, -49]\).

The synthetic division yields the quotient coefficients \([1, 1, 7]\) and a remainder of \(0\).

The quotient polynomial is formed from the coefficients \([1, 1, 7]\), which corresponds to: \[ x^2 + x + 7 \]

Final Answer