Division of Rational Expressions

Q. Let $F\left(x\right)=\frac{x}{(1+x^n{)}^{1/n}}$ for $n\ge 2$ and $g\left(x\right)=\underset{\text{f occours n times}}{\underset{}{(f\circ f\circ ..\circ f)}}\left(x\right)$. Then $\int x^{x-2}g\left(x\right)dx$ equals.

1. $\frac{1}{n(n-1)}(1+n{x}^n{)}^{1-\frac{1}{n}}+K$
   
2. $\frac{1}{n(n+1)}(1+n{x}^n)1+\frac{1}{n}+K$
   
3. $\frac{1}{n-1}(1+n{x}^n{)}^{1-\frac{1}{n}}+k$
   
4. $\frac{1}{n+1}(1+n{x}^n{)}^{1+\frac{1}{n}}+K$

Q.

x+2 is a factor of $x^3+8$

1. False

2. True

Q. Express the following rational expression to its lowest form.
$\frac{x^3-27}{x^2-9}$

1. $\frac{(x^2-9x+9)}{(x+3)}$
2. $\frac{(x^2+3x+9)}{(x+3)}$
3. $\frac{(x^2+3x+6)}{(2x+3)}$
4. None of the above

Q. Express $\frac{6x^2-21x-12}{54x^2+45x+9}$ in its lowest form.

1. $\frac{(2x-1)}{9(3x+1)}$
2. $\frac{(x-4)}{3(3x+1)}$
3. $\frac{(2x+1)}{3(3x+1)}$
4. $\frac{(x-4)}{3(2x+1)}$

Q. Evaluate $\frac{3x^2-x-4}{9x^2-16}÷\frac{4x^2-4}{3x^2-2x-1}$

1. $\frac{x+1}{4(x+4)}$
2. $\frac{3x+1}{4(3x+4)}$
3. $\frac{3x+5}{4(3x+5)}$
4. None of the above

Q. $\frac{x}{x-1}÷\frac{x^2}{x^2-1}$ = .

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

Q. Simplify $\frac{3x^2-x-4}{9x^2-16}÷\frac{4x^2-4}{3x^2-2x-1}$

1. $\frac{3x-1}{4(3x-4)}$
2. $\frac{3x-1}{(3x+4)}$
3. $\frac{3x+1}{4(3x+4)}$
4. None of the above

Q. Evaluate $\frac{2x^2+5x-3}{2x^2+7x-6}÷\frac{2x^2+5x-3}{2x^2-x-6}$

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

Q. Simplify $\frac{x^2+11x+24}{x^2-15x+56}÷\frac{x^2-x-12}{x^2-11x+28}.$

1. $\frac{(x+8)}{(x-8)}$
2. $\frac{(x-8)}{(x+8)}$
3. $\frac{(x-4)}{(x+4)}$
4. $\frac{(x+4)}{(x-4)}$

Q. Simplify $\frac{9x^2+12x+4}{4x^2-27x-7}÷\frac{12x^2+5x-2}{16x^2-1}.$

1. $\frac{(3x+2)}{(x+7)}$
2. $\frac{(3x-2)}{(x-7)}$
3. $\frac{(3x-2)}{(x+7)}$
4. $\frac{(3x+2)}{(x-7)}$

Q.

What is the product of $(x^2–9x)/(x+5)$ and $(3x+15)/(x–9)$?

1. 3x

2. 9x

3. 5/x

4. 3/x

Q. Evaluate $\frac{x}{x+1}÷\frac{x^2}{x^2-1}$

1. $\frac{x-1}{x}$
2. $\frac{x+1}{x}$
3. $\frac{2x-1}{x}$
4. None of the above

Q.

Factorise $\frac{12x^4+8x^3-2x}{2x}$.

Q. What is $\frac{x^2-3x+2}{x^2-5x+6}÷\frac{x^2-5x+4}{x^2-7x+12}$ equal to

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