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 407.
The remainder is 407.

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

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

The remainder is 2,087.
The remainder is 2,087.
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page 19 of 19

Answer 1

To find the remainder of the polynomial division \( \frac{3x^4 + 9x^2 - 13}{x^2 - 5x} \), we will perform the polynomial long division step by step.

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

2. First term of the quotient: To get the leading term, divide \( 3x^4 \) by \( x^2 \): \[ \frac{3x^4}{x^2} = 3x^2 \]

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

4. Next term of the quotient: Now divide \( 15x^3 \) by \( x^2 \): \[ \frac{15x^3}{x^2} = 15x \] Multiply and subtract: \[ 15x(x^2 - 5x) = 15x^3 - 75x^2 \] Now subtract: \[ (15x^3 + 9x^2 - 13) - (15x^3 - 75x^2) = 84x^2 - 13 \]

5. Next term of the quotient: Divide \( 84x^2 \) by \( x^2 \): \[ \frac{84x^2}{x^2} = 84 \] Multiply and subtract: \[ 84(x^2 - 5x) = 84x^2 - 420x \] Now subtract: \[ (84x^2 - 13) - (84x^2 - 420x) = 420x - 13 \]

At this point, we cannot divide anymore because the degree of the remainder \( 420x - 13 \) (degree 1) is less than the degree of the divisor \( x^2 - 5x \) (degree 2).

Thus, the remainder is:

\[ \text{Remainder} = 420x - 13 \]

Final Answer: The remainder is \( 420x - 13 \).