Question

4 | 2 -9 -21 88 48
↓ 8 -4 -100 -48
2 -1 -25 -12 0
what is the proper way to write the answer for the following problem?
options:
$2x^4 - x^3 - 25x^2 - 12x + 0$
$2x^4 - x^3 - 25x^2 - 12x - 0$
$2x^3 - x^2 - 25x - 12$
$2x^3 - x^2 - 25x + 12$

Explanation:

Step1: Identify original polynomial

The top row of coefficients $2, -9, -21, 88, 48$ corresponds to $2x^4 -9x^3 -21x^2 +88x +48$.

Step2: Interpret synthetic division result

Synthetic division with root $4$ gives bottom row coefficients $2, -1, -25, -12, 0$. The final $0$ confirms $x=4$ is a root, so we factor out $(x-4)$.

Step3: Form quotient polynomial

The first 4 coefficients represent the quotient: $2x^3 -1x^2 -25x -12$, since dividing a 4th-degree polynomial by linear term gives 3rd-degree polynomial.

Answer:

$2x^3 - x^2 -25x -12$