Question

1. divide $2x^3 - 1$ by $x - 2$.

Explanation:

Step1: Set up polynomial long division

Divide $2x^3 + 0x^2 + 0x - 1$ by $x - 2$

Step2: Divide leading terms

$\frac{2x^3}{x} = 2x^2$, multiply divisor: $2x^2(x-2)=2x^3-4x^2$
Subtract: $(2x^3+0x^2)-(2x^3-4x^2)=4x^2$

Step3: Bring down next term, repeat

$\frac{4x^2}{x}=4x$, multiply divisor: $4x(x-2)=4x^2-8x$
Subtract: $(4x^2+0x)-(4x^2-8x)=8x$

Step4: Bring down last term, repeat

$\frac{8x}{x}=8$, multiply divisor: $8(x-2)=8x-16$
Subtract: $(8x-1)-(8x-16)=15$

Answer:

Quotient: $2x^2 + 4x + 8$, Remainder: $15$, or written as $2x^2 + 4x + 8 + \frac{15}{x-2}$