Find the quotient (3x^2 - 4x - 1) divided by (x - 2).

Step-by-step solution

1. 1

   Set up the long division. Write the dividend $3x^2-4x-1$ in descending powers (it already is) and divide by $x-2$. At each stage you match only the leading term of what is left.

2. 2

   First division step. $\dfrac{3x^2}{x}=3x$, so the first quotient term is $3x$. Multiply back: $3x(x-2)=3x^2-6x$. Subtract: $(3x^2-4x)-(3x^2-6x)=2x$, and bring down the $-1$ to get $2x-1$.

3. 3

   Second division step. $\dfrac{2x}{x}=2$, so the next quotient term is $2$. Multiply back: $2(x-2)=2x-4$. Subtract: $(2x-1)-(2x-4)=3$.

4. 4

   Stop and read off the result. The leftover $3$ has degree $0$, which is lower than the degree $1$ of the divisor, so the division ends. The quotient is $3x+2$ and the remainder is $3$.

5. 5

   Write the mixed form. $\dfrac{3x^2-4x-1}{x-2}=3x+2+\dfrac{3}{x-2}$. Because the remainder is not zero, $x-2$ is not a factor of the dividend.

6. 6

   Confirm with the Remainder Theorem and by multiplying back. Substituting $x=2$ into the dividend gives $3(4)-8-1=3$, matching the remainder. Multiplying out, $(x-2)(3x+2)+3=3x^2+2x-6x-4+3=3x^2-4x-1$, the original dividend.