Question

The image shows the first few steps of the polynomial division (3x4+9x2−13)÷(x2−5x) . Complete the division process and find the remainder of this polynomial division. (1 point) Responses The remainder is 407x . The remainder is 407 x . The remainder is 2,087. The remainder is 2,087. The remainder is 407. The remainder is 407. The remainder is 420x−13 .

Answer 1

To complete the division process of \( (3x^4 + 9x^2 - 13) ÷ (x^2 - 5x) \) using polynomial long division, we will divide step by step.

1. **Set up the division:** Divide Polynomial \(3x^4 + 0x^3 + 9x^2 - 13\) by \(x^2 - 5x\).

2. **First term:** Divide \(3x^4\) by \(x^2\) to get \(3x^2\).
- Multiply \(3x^2\) by the divisor \(x^2 - 5x\):
\[
3x^2(x^2 - 5x) = 3x^4 - 15x^3
\]
- Subtract this from the original polynomial:
\[
(3x^4 + 0x^3 + 9x^2) - (3x^4 - 15x^3) = 15x^3 + 9x^2
\]

3. **Next term:** Divide \(15x^3\) by \(x^2\) to get \(15x\).
- Multiply \(15x\) by \(x^2 - 5x\):
\[
15x(x^2 - 5x) = 15x^3 - 75x^2
\]
- Subtract this from \(15x^3 + 9x^2\):
\[
(15x^3 + 9x^2) - (15x^3 - 75x^2) = 84x^2
\]

4. **Next term:** Divide \(84x^2\) by \(x^2\) to get \(84\).
- Multiply \(84\) by \(x^2 - 5x\):
\[
84(x^2 - 5x) = 84x^2 - 420x
\]
- Subtract this from \(84x^2\):
\[
(84x^2) - (84x^2 - 420x) = 420x
\]

5. **Since the degree of the remaining polynomial \(420x\) is less than the degree of the divisor \(x^2 - 5x\), we stop here.**

Therefore, the result of the division can be expressed as:

\[
3x^2 + 15x + 84 + \frac{420x}{x^2 - 5x}
\]

Thus, the remainder is:

\[
\text{Remainder } = 420x
\]

From the options provided, the correct answer is:

**The remainder is \(420x\).**