Q: Find the domain of the function $$R(x) = \dfrac{3 + 2x}{x^3$ + x^2$ - 2x}.$$

$$x \neq 0,\ x \neq 1,\ x \neq -2$$

Question 1

Which of the following is the correct domain of the function $$f(x) = \dfrac{x^2$}{x^2$ - 4}$$?

Accepted Answer

$$x 
eq 2,\ x 
eq -2$$

Question 2

Simplify the rational expression: $$\dfrac{x^2$ - 3x - 18}{x - 6}$$

Accepted Answer

$$x + 3$$

Question 3

A chemist is mixing two solutions. The concentration of the first solution is represented by $$\dfrac{3x^2$ + 2x - 4}{4x - 5}. $$For which value of $$x is $$this expression undefined?

Accepted Answer

$$x = \dfrac{5}{4}$$

Question 4

Which of the following is the correct simplification of $$\dfrac{2 - x}{x - 2}$$?

Accepted Answer

$$-1$$

Question 5

Multiply and simplify: $$\dfrac{1 + 3n}{2n} \cdot \dfrac{2n - 4}{3n^2$ - 2n - 1}$$

Accepted Answer

$$\dfrac{(1 + 3n)(n - 2)}{n(3n + 1)(n - 1)}$$

Question 6

Divide and simplify: $$\dfrac{8m^2$}{3m^2$ - 12} \div \dfrac{40}{2 - m}$$

Accepted Answer

$$-\dfrac{m^2$ (2 - m)}{15(m + 2)}$$

Question 7

Which of the following is NOT a rational expression?

Accepted Answer

$$x + 2$$

Question 8

A medical researcher uses the expression $$\dfrac{y - 5}{y^2$ + 4} to $$model a drug's effectiveness. For which value of $$y is $$this expression undefined?

Accepted Answer

$$	ext{None of these}$$

Question 9

Simplify the expression: $$\dfrac{4x^4$ - 4x^3$}{x^4$ - x^2$}$$

Accepted Answer

$$\dfrac{4x}{x - 1}$$

Question 10

Which of the following is a correct step when simplifying a rational expression?

Accepted Answer

Factor both numerator and denominator completely, then cancel common factors.

Question 11

Divide and simplify: $$\dfrac{a^2$ - a - 6}{a^2$ - 81} \div \dfrac{a^2$ - 7a - 18}{4a + 36}$$

Accepted Answer

$$\dfrac{(a - 3)(a + 2)(4a + 36)}{(a - 9)(a + 9)(a^2$ - 7a - 18)}$$

Question 12

Which of the following is simplified correctly?

Accepted Answer

$$\dfrac{3x}{3x - 3} = \dfrac{x}{x - 1}$$

Question 13

Find the domain of the function $$R(x) = \dfrac{3 + 2x}{x^3$ + x^2$ - 2x}.$$

Accepted Answer

$$x 
eq 0,\ x 
eq 1,\ x 
eq -2$$

Question 14

A biologist uses the expression $$\dfrac{xy - y^2$}{3y} to $$model a population ratio. Simplify this expression.

Accepted Answer

$$\dfrac{x - y}{3}$$

Question 15

Multiply and simplify: $$\dfrac{4a^2$ - 8a}{ab - 2b + 3a - 6} \cdot \dfrac{8b + 24}{3a + 6}$$

Accepted Answer

$$\dfrac{4a(a - 2)(8b + 24)}{(a + 3)(b - 2)(3a + 6)}$$

Question 16

Which of the following steps is NOT valid when multiplying rational expressions?

Accepted Answer

Cancel terms that are not factors.

Question 17

Divide and simplify: $$\dfrac{5x^2$ - 13x - 6}{9x^2$ + 17x - 2} \div \dfrac{8x^2$ - 23x - 3}{8x^2$ + 17x + 2}$$

Accepted Answer

$$\dfrac{(5x^2$ - 13x - 6)(8x^2$ + 17x + 2)}{(9x^2$ + 17x - 2)(8x^2$ - 23x - 3)}$$

Rational Functions and Operations on Rational Expressions

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