A rational expression is the quotient of two polynomials, where the denominator is not equal to zero. When we first work with rational expressions, we encounter two tasks: find the restricted values and simplify. We find the restricted values by identifying what values create a denominator of zero. We simplify a rational expression by first factoring the numerator and denominator and then canceling common factors.

Test Objectives
• Demonstrate the ability to find the domain of a rational expression
• Demonstrate the ability to simplify a rational expression
Simplifying Rational Expressions Practice Test:

#1:

Instructions: Simplify each, state the domain.

$$a)\hspace{.2em}\frac{15x}{10x - 10}$$

$$b)\hspace{.2em}\frac{x + 3}{x^2 + 4x + 3}$$

#2:

Instructions: Simplify each, state the domain.

$$a)\hspace{.2em}\frac{x^2 + 15x + 54}{10x^2 + 60x}$$

$$b)\hspace{.2em}\frac{x^2 - 12x + 20}{x^2 - 19x + 90}$$

#3:

Instructions: Simplify each, state the domain.

$$a)\hspace{.2em}\frac{x^2 + 4x - 60}{9x + 72}$$

$$b)\hspace{.2em}\frac{6x^2 + 18x - 24}{21x^2 - 33x + 12}$$

#4:

Instructions: Simplify each, state the domain.

$$a)\hspace{.2em}\frac{5x - 45}{-10x^3 + 100x^2 - 90x}$$

$$b)\hspace{.2em}\frac{4x^3 - 38x^2 - 20x}{14x^2 - 146x + 60}$$

#5:

Instructions: Simplify each, state the domain.

$$a)\hspace{.2em}\frac{10x^3 + 70x^2 - 80x}{2x + 16}$$

$$b)\hspace{.2em}\frac{56x^2 + 120x + 16}{16x^2 - 24x - 112}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}\frac{3x}{2(x - 1)}$$ $$\left\{x | x ≠ 1\right\}$$

$$b)\hspace{.2em}\frac{1}{x + 1}$$ $$\left\{x | x ≠ -3, -1\right\}$$

#2:

Solutions:

$$a)\hspace{.2em}\frac{x + 9}{10x}$$ $$\left\{x | x ≠ 0, -6\right\}$$

$$b)\hspace{.2em}\frac{x - 2}{x - 9}$$ $$\left\{x | x ≠ 9, 10\right\}$$

#3:

Solutions:

$$a)\hspace{.2em}\frac{(x - 6)(x + 10)}{9(x + 8)}$$ $$\left\{x | x ≠ -8\right\}$$

$$b)\hspace{.2em}\frac{2(x + 4)}{7x - 4}$$ $$\left\{x | x ≠ 1, \frac{4}{7}\right\}$$

#4:

Solutions:

$$a)\hspace{.2em}\frac{1}{2x(-x + 1)}$$ $$\left\{x | x ≠ 0, 1, 9 \right\}$$

$$b)\hspace{.2em}\frac{x(2x + 1)}{7x - 3}$$ $$\left\{x | x ≠ \frac{3}{7}, 10\right\}$$

#5:

Solutions:

$$a)\hspace{.2em}5x(x - 1)$$ $$\left\{x | x ≠ -8\right\}$$

$$b)\hspace{.2em}\frac{7x + 1}{2x - 7}$$ $$\left\{x | x ≠ -2, \frac{7}{2}\right\}$$