Question

divide the polynomial (2x^3 - 5x^2 - 4x - 25) by (x - 4). the options are: (2x^3 + 3x + (7/(x - 4))), (2x^2 + 3x + 8 + (7/(x - 4))), (2x^2 - x + 10 + (6/(x - 4))), (2x^2 + 3x + 8)

Explanation:

Step1: Polynomial long division setup

Divide $2x^3 -5x^2 -4x -25$ by $x-4$

Step2: Divide leading terms

$\frac{2x^3}{x}=2x^2$. Multiply $x-4$ by $2x^2$: $2x^3-8x^2$

Step3: Subtract from dividend

$(2x^3-5x^2-4x-25)-(2x^3-8x^2)=3x^2-4x-25$

Step4: Divide new leading terms

$\frac{3x^2}{x}=3x$. Multiply $x-4$ by $3x$: $3x^2-12x$

Step5: Subtract from current remainder

$(3x^2-4x-25)-(3x^2-12x)=8x-25$

Step6: Divide leading terms again

$\frac{8x}{x}=8$. Multiply $x-4$ by $8$: $8x-32$

Step7: Find final remainder

$(8x-25)-(8x-32)=7$

Step8: Combine results

Quotient: $2x^2+3x+8$, Remainder: $7$. So $\frac{2x^3-5x^2-4x-25}{x-4}=2x^2+3x+8+\frac{7}{x-4}$

Answer:

B. $2x^{2}+3x+8+\frac{7}{x-4}$