Question

(3x³ + 5x - 1) ÷ (x + 1)
options:
3x² + 3x + 8 + 7/(x+1)
3x³ + 3x² + 8x + 7
3x² - 3x + 8 - 9/(x+1)
3x³ - 3x² + 8x - 9

Explanation:

Step1: Set up polynomial long division

Divide $3x^3 + 0x^2 + 5x - 1$ by $x+1$ (add $0x^2$ for missing term)

Step2: Divide leading terms

$\frac{3x^3}{x} = 3x^2$. Multiply divisor: $3x^2(x+1)=3x^3+3x^2$
Subtract: $(3x^3+0x^2)-(3x^3+3x^2)=-3x^2$
Bring down $5x$: $-3x^2+5x$

Step3: Divide leading terms again

$\frac{-3x^2}{x}=-3x$. Multiply divisor: $-3x(x+1)=-3x^2-3x$
Subtract: $(-3x^2+5x)-(-3x^2-3x)=8x$
Bring down $-1$: $8x-1$

Step4: Divide leading terms a third time

$\frac{8x}{x}=8$. Multiply divisor: $8(x+1)=8x+8$
Subtract: $(8x-1)-(8x+8)=-9$

Step5: Combine results

Quotient: $3x^2-3x+8$, remainder: $-9$. Write as $3x^2 - 3x + 8 - \frac{9}{x+1}$

Answer:

$3x^2 - 3x + 8 - \frac{9}{x+1}$ (third option)