Question

the right side has the expression \\(\frac{3x^2 + 5x - 1}{x + 1}\\), and the left side has four colored cards with expressions:

• yellow card: \\(3x^3 - 3x^2 + 8x - 9\\)
• purple card: \\(3x^2 - 3x + 8 - \frac{9}{x + 1}\\)
• orange card: \\(3x^2 + 3x + 8 + \frac{7}{x + 1}\\)
• cyan card: \\(3x^3 + 3x^2 + 8x + 7\\)

Explanation:

Step1: Set up polynomial division

We need to compute $\frac{3x^3 + 5x - 1}{x+1}$. Use polynomial long division or synthetic division. For synthetic division, root of $x+1=0$ is $x=-1$.
Coefficients of dividend: $3$ (for $x^3$), $0$ (for $x^2$), $5$ (for $x$), $-1$ (constant).

Step2: Perform synthetic division

Bring down the 3. Multiply by -1: $3\times(-1)=-3$. Add to next coefficient: $0+(-3)=-3$.
Multiply -3 by -1: $(-3)\times(-1)=3$. Add to next coefficient: $5+3=8$.
Multiply 8 by -1: $8\times(-1)=-8$. Add to last coefficient: $-1+(-8)=-9$.

Step3: Write quotient and remainder

Quotient: $3x^2 - 3x + 8$, remainder: $-9$. So the result is $3x^2 - 3x + 8 - \frac{9}{x+1}$.

Answer:

$3x^2 - 3x + 8 - \frac{9}{x+1}$