Questions: Divide using synthetic division. (x^4+625) ÷ (x-5)

Solution Steps

We begin by dividing the polynomial \(x^{4} + 625\) by the linear polynomial \(x - 5\).

1. Divide \(x^{4}\) by \(x\), resulting in \(x^{3}\). The remainder is \(5x^{3} + 625\).
2. Next, divide \(5x^{3}\) by \(x\), yielding \(5x^{2}\) with a remainder of \(25x^{2} + 625\).
3. Then, divide \(25x^{2}\) by \(x\), which gives \(25x\) and leaves a remainder of \(125x + 625\).
4. Finally, divide \(125x\) by \(x\), resulting in \(125\) with a remainder of \(1250\).

From the division process, we can compile the results:

• The quotient is \(x^{3} + 5x^{2} + 25x + 125\).
• The remainder is \(1250\).

We can express the result of the division in the following form: \[ \frac{x^{4} + 625}{x - 5} = x^{3} + 5x^{2} + 25x + 125 + \frac{1250}{x - 5} \]

Final Answer