### About Rational Expressions:

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 solve a quadratic equation by factoring
- Demonstrate the ability to find the restricted values for a rational expression
- Demonstrate the ability to simplify a rational expression

#1:

Instructions: Find the domain.

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

b) $$h(x) = \frac{x^2 + 2x - 8}{4x - 7}$$

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#2:

Instructions: Find the domain.

a) $$f(x) = \frac{x^2 - x}{-2x^3 + 4x^2 + 6x}$$

b) $$h(x) = \frac{6x^4 - 66x^3 + 60x^2}{4x^3 - 36x}$$

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#3:

Instructions: Simplify each.

a) $$\frac{r - 1}{r^2 - 4r + 3}$$

b) $$\frac{3m - 9}{3 - m}$$

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#4:

Instructions: Simplify each.

a) $$\frac{5k + 35}{10k^2 + 60k - 70}$$

b) $$\frac{n^3 + 4n^2 - 32n}{n^2 + 12n + 32}$$

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#5:

Instructions: Simplify each.

a) $$\frac{5n^2 + 15n + 10}{-n^2 - 3n - 2}$$

b) $$\frac{5x^2 + 6x + 1}{6x^2 + 8x + 2}$$

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Written Solutions:

#1:

Solutions:

a) $$\left\{x|x≠ -1\right\}$$

b) $$\left\{x|x≠ \frac{7}{4}\right\}$$

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#2:

Solutions:

a) $$\left\{x|x≠ -1,0,3\right\}$$

b) $$\left\{x|x≠ -3,0,3\right\}$$

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#3:

Solutions:

a) $$\frac{1}{r - 3}$$

b) $$-3$$

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#4:

Solutions:

a) $$\frac{1}{2(k - 1)}$$

b) $$\frac{n(n - 4)}{n + 4}$$

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#5:

Solutions:

a) $$-5$$

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