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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 2,087.
The remainder is 2,087.

The remainder is 407.
The remainder is 407.

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

The remainder is 407x
.

1 answer

To complete the polynomial division of \((3x^4 + 9x^2 - 13)\) by \((x^2 - 5x)\), we can go through the steps involved in polynomial long division.

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

  2. Multiply and subtract: Multiply \(3x^2\) by the whole divisor \((x^2 - 5x)\) and subtract from the dividend: \[ 3x^2 \cdot (x^2 - 5x) = 3x^4 - 15x^3 \] Subtracting: \[ (3x^4 + 9x^2 - 13) - (3x^4 - 15x^3) = 15x^3 + 9x^2 - 13 \]

  3. Repeat the process: Now take the result \(15x^3 + 9x^2 - 13\) and repeat the division: \[ 15x^3 \div x^2 = 15x \] Multiply and subtract: \[ 15x \cdot (x^2 - 5x) = 15x^3 - 75x^2 \] Subtracting: \[ (15x^3 + 9x^2 - 13) - (15x^3 - 75x^2) = 84x^2 - 13 \]

  4. Continue: Now divide \(84x^2 - 13\): \[ 84x^2 \div x^2 = 84 \] Multiply and subtract again: \[ 84 \cdot (x^2 - 5x) = 84x^2 - 420x \] Subtracting: \[ (84x^2 - 13) - (84x^2 - 420x) = 420x - 13 \]

So far, we have:

  • The quotient is \(3x^2 + 15x + 84\).
  • The remainder is \(420x - 13\).

Thus, the answer is: The remainder is \(420x - 13\).