Factorization & Simplification — Quiz 5

Questions in this quiz (10)

1. Given that $$x + 3y = 3$$, what is the value of the following expression?$$\frac{x}{y - 1} + \left( \frac{x}{y - 1} \right)^2 + \left( \frac{x}{y - 1} \right)^3$$

   • 39
   • 21
   • -21
   • -15

2. Simplify the following expression, assuming ($$x \neq 0$$ and $$y \neq 0$$):$$\frac{(x^3)^4 \cdot (x^{-2})^4 \cdot \sqrt{y^3}}{x^4 \cdot x^{-2} \cdot y^{1/2}}$$

   • $$x^2 \cdot y^3$$
   • $$x^3 \cdot y^2$$
   • $$x^2 \cdot y$$
   • $$x \cdot y^2$$

3. The expression $$n!$$ (read as $$n$$ factorial) is defined as the product of all positive integers up to and including $$n$$, whenever $$n$$ is a positive integer. For example, $$4!=1\cdot2\cdot3\cdot4$$Whenever $$n$$ is a positive integer, which of the following is equivalent to $$\frac{(n+2)! \cdot 5!}{n! \cdot 4!}$$?

   • $$5(n+1)(n+2)$$
   • $$\frac{5(n+2)}{4}$$
   • $$\frac{2(n+2)}{n}$$
   • $$\frac{5(n+1)(n+2)}{4}$$

4. Simplify the expression: $$\frac{\frac{c}{b}-b}{\frac{1}{b^{-1}}}$$.

   • $$c-b^2$$
   • $$\frac{c}{b} - 1$$
   • $$\frac{c}{b^2} - 1$$
   • $$\frac{c}{b^2} - b$$

5. Simplify the given expression:$$\frac{(4x^2+7x-2)(6x^3-24x^2-18x)}{(x+2)(2x^2-8x-10)}$$

   • $$\frac{(x-1)(x-3)}{(x)}$$
   • $$\frac{3x(4x-1)}{(x-3)}$$
   • $$\frac{3x(4x-1)(x-3)}{(x-5)}$$
   • $$\frac{3x(4x-3)}{(x-4)}$$

6. Simplify the expression: $$\frac{1}{\sqrt{n+3}+\sqrt{n}}$$

   • $$\frac{\sqrt{n+3}-\sqrt{n}}{3}$$
   • $$\frac{\sqrt{n+3}}{3}$$
   • $$\frac{-\sqrt{n}}{3}$$
   • $$\frac{\sqrt{n}}{3}$$

7. Simplify the expression:$$\frac{1+\sqrt{y}}{1-\sqrt{y}}+\frac{\sqrt{y}-1}{\sqrt{y}}$$

   • $$\frac{2\sqrt{y}+1}{y\sqrt{y}}$$
   • $$\frac{\sqrt{y}+3y}{(1-y)\sqrt{1-y}}$$
   • $$\frac{2\sqrt{y}+3y -1}{-y\sqrt{y}}$$
   • $$\frac{2\sqrt{y}+3y -1}{(1-y)\sqrt{y}}$$

8. The expression $$45x^2 - 54xy + 16y^2$$ is equivalent to:

   • $$(15x - 8y)(3x - 2y)$$
   • $$(45x - 3y)(x - 2y)$$
   • $$(45x + 3y)(4x - 2y)$$
   • $$(5x - 3y)(4x + 2y)$$

9. The expression $$\frac{3x^2 + 2x - 5}{x^2 - 4}$$ is equivalent to:

   • $$\frac{(x + 1)}{(x + 2)}$$
   • $$\frac{(2x - 1)}{(x + 3)}$$
   • $$\frac{(3x + 1)}{(2x - 1)}$$
   • $$\frac{(x - 1)(3x + 5)}{(x - 2)(x + 2)}$$

10. The expression $$\frac{72x^2y^3z^4 - 24x^3y^2z}{6x^2y^2z^2}$$ can be simplified to:

    • $$\frac{12(3y z^2 - x)}{z}$$
    • $$\frac{4(3y z^3 - x)}{z}$$
    • $$\frac{4(2y z^2 + x)}{z}$$
    • $$\frac{12(y z^2 + x)}{z}$$