Question

determine the quotient and remainder.\\((4x^3 + x - 2) \div (x - 2)\\)

Explanation:

Step1: Set up polynomial long division

Divide $4x^3 + 0x^2 + x - 2$ by $x-2$ (insert $0x^2$ for missing term)

Step2: Divide leading terms

$\frac{4x^3}{x} = 4x^2$, multiply divisor: $4x^2(x-2)=4x^3-8x^2$
Subtract: $(4x^3+0x^2)-(4x^3-8x^2)=8x^2$
Bring down $x$: $8x^2 + x$

Step3: Divide new leading terms

$\frac{8x^2}{x}=8x$, multiply divisor: $8x(x-2)=8x^2-16x$
Subtract: $(8x^2+x)-(8x^2-16x)=17x$
Bring down $-2$: $17x - 2$

Step4: Divide new leading terms

$\frac{17x}{x}=17$, multiply divisor: $17(x-2)=17x-34$
Subtract: $(17x-2)-(17x-34)=32$

Answer:

Quotient: $4x^2 + 8x + 17$
Remainder: $32$