Question

simplify the rational expression \\(\frac{6x^3 + 5x^2 - 4x - 4}{2x - 1}\\) to obtain a quotient and a remainder. use the keyboard to write your answer in the box. you may use geogebra cas calculator or any cas calculator. the simplified form of the rational expression \\(\frac{6x^3 + 5x^2 - 4x - 4}{2x - 1}\\) is \\(square\\).

Explanation:

Step1: Divide leading terms

$\frac{6x^3}{2x} = 3x^2$
Multiply divisor by $3x^2$: $3x^2(2x-1)=6x^3-3x^2$
Subtract from dividend:
$(6x^3+5x^2-4x-4)-(6x^3-3x^2)=8x^2-4x-4$

Step2: Divide new leading terms

$\frac{8x^2}{2x}=4x$
Multiply divisor by $4x$: $4x(2x-1)=8x^2-4x$
Subtract from new dividend:
$(8x^2-4x-4)-(8x^2-4x)=-4$

Step3: Write final form

Quotient is $3x^2+4x$, remainder is $-4$.
Expression: $\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$

Answer:

$3x^2 + 4x - \frac{4}{2x-1}$