Use the image to answer the question.

Question

A long division has for its divisor 3 x superscript 4 baseline plus 9 x squared minus 13. The dividend is x squared minus 5 x.

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 2,087.
The remainder is 2,087.

The remainder is 407x.
The remainder is 407 x .

The remainder is 420x−13.
The remainder is 420 x minus 13 .

The remainder is 407.

Answer 1

To perform the polynomial long division of \( (3x^4 + 9x^2 - 13) \) by \( (x^2 - 5x) \), follow these steps:

1. Divide the leading term: Divide the leading term of the dividend \( 3x^4 \) by the leading term of the divisor \( x^2 \): \[ \frac{3x^4}{x^2} = 3x^2 \]

2. Multiply and subtract: Multiply \( 3x^2 \) by the entire divisor \( x^2 - 5x \): \[ 3x^2(x^2 - 5x) = 3x^4 - 15x^3 \] Subtract this product from the original polynomial: \[ (3x^4 + 9x^2 - 13) - (3x^4 - 15x^3) = 15x^3 + 9x^2 - 13 \]

3. Repeat the process: Now, take the leading term \( 15x^3 \) and divide by the leading term of the divisor \( x^2 \): \[ \frac{15x^3}{x^2} = 15x \] Multiply and subtract again: \[ 15x(x^2 - 5x) = 15x^3 - 75x^2 \] Subtract: \[ (15x^3 + 9x^2 - 13) - (15x^3 - 75x^2) = 84x^2 - 13 \]

4. Continue: Take the leading term \( 84x^2 \) and divide: \[ \frac{84x^2}{x^2} = 84 \] Multiply and subtract: \[ 84(x^2 - 5x) = 84x^2 - 420x \] Subtract: \[ (84x^2 - 13) - (84x^2 - 420x) = 420x - 13 \]

Now we cannot divide further since the degree of \( 420x - 13 \) (which is 1) is less than the degree of the divisor \( x^2 - 5x \) (which is 2).

Thus, the final quotient is \( 3x^2 + 15x + 84 \) and the remainder is \( 420x - 13 \).

Hence, the remainder of this polynomial division is: \[ \text{The remainder is } 420x - 13. \]