Question

1. simplify the following rational expression: $\frac{x^2}{y^2}+\frac{x}{y^3}$

a. $\frac{x^3 + x}{y^3}$
b. $\frac{x^2 + xy}{y^3}$
c. $\frac{x^2y + xy}{y^3}$
d. $\frac{x^2y + x}{y^3}$

• 2. which of the following represents the factored form of the expression $6x^2 - 7x - 3$?

a. $(3x - 2)(2x - 1)$
b. $(2x - 3)(3x + 1)$
c. $(2x + 3)(3x - 1)$
d. $(3x + 1)(2x + 3)$

1. which of the following expressions is equivalent to $(a + b)(a - b)$?

a. $a^2 - b^2$
b. $(a - b)^2$
c. $ab(a - b)$
d. $a(b - a)$

Explanation:

Step1: Simplify question 1, get common denominator

The common denominator of $\frac{x^2}{y^2}$ and $\frac{x}{y^3}$ is $y^3$. Rewrite the first term:
$\frac{x^2}{y^2} = \frac{x^2 \cdot y}{y^2 \cdot y} = \frac{x^2 y}{y^3}$

Step2: Add the two rational terms

Add the fractions with the common denominator:
$\frac{x^2 y}{y^3} + \frac{x}{y^3} = \frac{x^2 y + x}{y^3}$

Step3: Factor quadratic for question 2

For $6x^2 -7x -3$, use factoring by grouping. Split $-7x$ into $2x-9x$:
$6x^2 +2x -9x -3 = 2x(3x+1) -3(3x+1) = (3x+1)(2x-3)$

Step4: Expand $(a+b)(a-b)$ for question3

Use the difference of squares identity:
$(a+b)(a-b) = a^2 - b^2$

Answer:

1. d. $\frac{x^2 y + x}{y^3}$
2. d. $(3x + 1)(2x + 3)$
3. a. $a^2 - b^2$