Questions: (3x^3 + x + 5) / (x + 1)

Solution Steps

We start by dividing the polynomial \(3 x^{3} + x + 5\) by \(x + 1\).

1. Divide \(3 x^{3}\) by \(x\) to get \(3 x^{2}\).
2. Multiply \(3 x^{2}\) by \(x + 1\) to get \(3 x^{3} + 3 x^{2}\).
3. Subtract this from the original polynomial, resulting in \(-3 x^{2} + x + 5\).

Next, we continue the division with the new polynomial \(-3 x^{2} + x + 5\).

1. Divide \(-3 x^{2}\) by \(x\) to get \(-3 x\).
2. Multiply \(-3 x\) by \(x + 1\) to get \(-3 x^{2} - 3 x\).
3. Subtract this from \(-3 x^{2} + x + 5\), resulting in \(4 x + 5\).

We now divide \(4 x + 5\) by \(x + 1\).

1. Divide \(4 x\) by \(x\) to get \(4\).
2. Multiply \(4\) by \(x + 1\) to get \(4 x + 4\).
3. Subtract this from \(4 x + 5\), resulting in a remainder of \(1\).

Combining the results from the division, we have:

\[ \frac{3 x^{3} + x + 5}{x + 1} = 3 x^{2} - 3 x + 4 + \frac{1}{x + 1} \]

Thus, the quotient is \(3 x^{2} - 3 x + 4\) and the remainder is \(1\).

Final Answer