About Simplifying 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 find the domain of a rational expression
- Demonstrate the ability to simplify a rational expression
#1:
Instructions: Simplify each, state the domain.
$$a)\hspace{.2em}\frac{15x}{10x - 10}$$
$$b)\hspace{.2em}\frac{x + 3}{x^2 + 4x + 3}$$
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#2:
Instructions: Simplify each, state the domain.
$$a)\hspace{.2em}\frac{x^2 + 15x + 54}{10x^2 + 60x}$$
$$b)\hspace{.2em}\frac{x^2 - 12x + 20}{x^2 - 19x + 90}$$
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#3:
Instructions: Simplify each, state the domain.
$$a)\hspace{.2em}\frac{x^2 + 4x - 60}{9x + 72}$$
$$b)\hspace{.2em}\frac{6x^2 + 18x - 24}{21x^2 - 33x + 12}$$
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#4:
Instructions: Simplify each, state the domain.
$$a)\hspace{.2em}\frac{5x - 45}{-10x^3 + 100x^2 - 90x}$$
$$b)\hspace{.2em}\frac{4x^3 - 38x^2 - 20x}{14x^2 - 146x + 60}$$
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#5:
Instructions: Simplify each, state the domain.
$$a)\hspace{.2em}\frac{10x^3 + 70x^2 - 80x}{2x + 16}$$
$$b)\hspace{.2em}\frac{56x^2 + 120x + 16}{16x^2 - 24x - 112}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}\frac{3x}{2(x - 1)}$$ $$\left\{x | x ≠ 1\right\}$$
$$b)\hspace{.2em}\frac{1}{x + 1}$$ $$\left\{x | x ≠ -3, -1\right\}$$
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#2:
Solutions:
$$a)\hspace{.2em}\frac{x + 9}{10x}$$ $$\left\{x | x ≠ 0, -6\right\}$$
$$b)\hspace{.2em}\frac{x - 2}{x - 9}$$ $$\left\{x | x ≠ 9, 10\right\}$$
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#3:
Solutions:
$$a)\hspace{.2em}\frac{(x - 6)(x + 10)}{9(x + 8)}$$ $$\left\{x | x ≠ -8\right\}$$
$$b)\hspace{.2em}\frac{2(x + 4)}{7x - 4}$$ $$\left\{x | x ≠ 1, \frac{4}{7}\right\}$$
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#4:
Solutions:
$$a)\hspace{.2em}\frac{1}{2x(-x + 1)}$$ $$\left\{x | x ≠ 0, 1, 9 \right\}$$
$$b)\hspace{.2em}\frac{x(2x + 1)}{7x - 3}$$ $$\left\{x | x ≠ \frac{3}{7}, 10\right\}$$
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#5:
Solutions:
$$a)\hspace{.2em}5x(x - 1)$$ $$\left\{x | x ≠ -8\right\}$$
$$b)\hspace{.2em}\frac{7x + 1}{2x - 7}$$ $$\left\{x | x ≠ -2, \frac{7}{2}\right\}$$