Question

the expression on the right is (2x^3 - 5x^2 + 3x + 7div x - 2), and the options on the left (from top to bottom) are: (2x^3 - x^2 + x + 9), (2x^2 - x + 1), (2x^2 - x + 1 + \frac{9}{x - 2}), (2x^2 - 9x - 15 - \frac{23}{x - 2})

Explanation:

Step1: Set up polynomial division

We divide $\frac{2x^3 - 5x^2 + 3x + 7}{x - 2}$ using polynomial long division.
First, divide the leading term of the numerator by the leading term of the denominator: $\frac{2x^3}{x} = 2x^2$.
Multiply $x-2$ by $2x^2$: $2x^2(x-2) = 2x^3 - 4x^2$.
Subtract this from the numerator:
$$(2x^3 - 5x^2 + 3x + 7) - (2x^3 - 4x^2) = -x^2 + 3x + 7$$

Step2: Divide next leading term

Divide the new leading term $\frac{-x^2}{x} = -x$.
Multiply $x-2$ by $-x$: $-x(x-2) = -x^2 + 2x$.
Subtract this from the current remainder:
$$(-x^2 + 3x + 7) - (-x^2 + 2x) = x + 7$$

Step3: Divide final leading term

Divide the new leading term $\frac{x}{x} = 1$.
Multiply $x-2$ by $1$: $1(x-2) = x - 2$.
Subtract this from the current remainder:
$$(x + 7) - (x - 2) = 9$$

Step4: Write final expression

Combine the quotient terms and the remainder over the divisor:
$2x^2 - x + 1 + \frac{9}{x-2}$

Answer:

$2x^2 - x + 1 + \frac{9}{x-2}$