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 \(420x\).