Question

simplify the rational expression \\(\frac{x^2 - 1}{x - 1}\\) using long division method

fill in the blanks to complete the division:

the quotient is \\(\square\\) and the remainder is \\(\square\\).

Explanation:

Step1: Divide leading terms

$\frac{x^4}{x} = x^3$
Multiply divisor by $x^3$: $x^3(x-1) = x^4 - x^3$
Subtract from dividend: $(x^4 - 1) - (x^4 - x^3) = x^3 - 1$

Step2: Divide new leading terms

$\frac{x^3}{x} = x^2$
Multiply divisor by $x^2$: $x^2(x-1) = x^3 - x^2$
Subtract: $(x^3 - 1) - (x^3 - x^2) = x^2 - 1$

Step3: Divide new leading terms

$\frac{x^2}{x} = x$
Multiply divisor by $x$: $x(x-1) = x^2 - x$
Subtract: $(x^2 - 1) - (x^2 - x) = x - 1$

Step4: Divide final leading terms

$\frac{x}{x} = 1$
Multiply divisor by $1$: $1(x-1) = x - 1$
Subtract: $(x-1) - (x-1) = 0$

Answer:

The quotient is $x^3 + x^2 + x + 1$ and the remainder is $0$.