Question 1

Simplify the following expression and choose the correct answer:

$$\frac{r-s}{r+s} \cdot \frac{r^2$-s^2$$}{$$(r-s)^2$$}$

Accepted Answer

Question 2

Given $$\frac{m^2$-n^2$$}{r+s} \div \frac{m-n}{r+s}$, simplify the expression.

Accepted Answer

$$m+n$$

Question 3

Simplify the following product:

$$\frac{4x^2$+9x+2}{x^2$+x-2} \cdot \frac{x^2$-1}{3x^2$+x-2}$$

Accepted Answer

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

Question 4

A chemical reaction rate is modeled by $$\frac{a^2$-a-2}{a^2$-a-6} \div \frac{a^2$-2a}{2a+a^2$}. $$Simplify the expression to predict the rate for $$a \u003e 2.$$

Accepted Answer

$$\frac{a-2}{a}$$

Question 5

A company’s profit function is $$P(x) = \frac{3x+12}{2x-8} \div \frac{(x+4)^2$}{(x-4)^2$}. $$Simplify $$P(x).$$

Accepted Answer

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

Question 6

Simplify the following expression:

$$\frac{x^2$-y^2$$}{$$x^3-y^3$} \cdot \frac{x^2$+xy+y^2$$}{$$x^2+2xy+y^2$}$$

Accepted Answer

$$\frac{1}{x+y}$$

Question 7

If $$x$$ and $$y$$ are real numbers such that $$x^2+xy+y^2$=0$$, what can you conclude about $$x$$ and $$y$?

Accepted Answer

$$x = 0$$ and $$y = 0$$

Question 8

Which of the following is the correct factorization of $$x^2+x-2$?

Accepted Answer

$$(x+2)(x-1)$$

Question 9

Given $$a^2-2a$, which of the following is a correct factorization?

Accepted Answer

$$a(a-2)$$

Question 10

A population model uses $$x^3-y^3$ in the denominator. For which values of x$ and y$ is the expression undefined?

Accepted Answer

$$x = y$$

Question 11

Which of the following is equivalent to $$\frac{r^2$-s^2$$}{$$(r-s)^2$$}$?

Accepted Answer

$$\frac{(r+s)(r-s)}{(r-s)^2$}$$

Question 12

If $$x^2-1$ is in the numerator, which of the following is its factorization?

Accepted Answer

$$(x+1)(x-1)$$

Question 13

Which of the following is the correct simplification for $$\frac{a^2$-a-2}{a^2$-a-6}$$?

Accepted Answer

$$\frac{a-2}{a+2}$$

Question 14

A physics formula involves $$x^2+2xy+y^2$ in the denominator. Which of the following is an equivalent expression?

Accepted Answer

$$(x+y)^2$$

Question 15

Which of the following is the correct simplification for $$\frac{3x+12}{2x-8}$$?

Accepted Answer

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

Question 16

If $$x^2-y^2$ is in the numerator, which of the following is its factorization?

Accepted Answer

$$(x+y)(x-y)$$

Question 17

Which of the following is the correct simplification for $$\frac{a^2$-2a}{2a+a^2$}$$?

Accepted Answer

$$\frac{a(a-2)}{a(2+a)}$$

Question 18

A real-world scenario involves dividing $$x^2+xy+y^2$ by x^2$+2xy+y^2. $$What is the simplified form?

Accepted Answer

$$\frac{x^2$+xy+y^2$$}{$$(x+y)^2$$}$

Question 19

Which of the following is the correct simplification for $$\frac{(x+4)^2$}{(x-4)^2$}$$?

Accepted Answer

$$\left(\frac{x+4}{x-4}\$$right)^2$$

Rational Expressions and Operations: Precalculus Study Notes

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