Question

Use the image to answer the 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 \( 420x - 13 \).