what is the quotient of (3x^4 - 4x^2 + 8x - 1)÷(x - 2)?\n3x^3 + 6x^2 + 8x + 24 - \\frac{47}{x - 2}\n3x^3 +…

Answer:

C. $3x^{3}+6x^{2}+8x + 24+\frac{47}{x - 2}$

Explanation:

Step1: Use polynomial long - division

We divide $3x^{4}-4x^{2}+8x - 1$ by $x - 2$. First, divide the leading term of the dividend $3x^{4}$ by the leading term of the divisor $x$. We get $3x^{3}$. Multiply $x - 2$ by $3x^{3}$ to get $3x^{4}-6x^{3}$. Subtract this from the dividend: $(3x^{4}-4x^{2}+8x - 1)-(3x^{4}-6x^{3})=6x^{3}-4x^{2}+8x - 1$.

Step2: Continue the division

Divide the leading term of the new dividend $6x^{3}$ by $x$ to get $6x^{2}$. Multiply $x - 2$ by $6x^{2}$ to get $6x^{3}-12x^{2}$. Subtract this from the new dividend: $(6x^{3}-4x^{2}+8x - 1)-(6x^{3}-12x^{2}) = 8x^{2}+8x - 1$.

Step3: Keep dividing

Divide the leading term of the new dividend $8x^{2}$ by $x$ to get $8x$. Multiply $x - 2$ by $8x$ to get $8x^{2}-16x$. Subtract this from the new dividend: $(8x^{2}+8x - 1)-(8x^{2}-16x)=24x - 1$.

Step4: Final division step

Divide the leading term of the new dividend $24x$ by $x$ to get $24$. Multiply $x - 2$ by $24$ to get $24x-48$. Subtract this from the new dividend: $(24x - 1)-(24x - 48)=47$. So the quotient is $3x^{3}+6x^{2}+8x + 24$ and the remainder is $47$, and the result is $3x^{3}+6x^{2}+8x + 24+\frac{47}{x - 2}$.