Advanced Algebra - Rational Expressions

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What makes a rational expression undefined?

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A rational expression is undefined when its denominator equals zero. For example, the expression \( \frac{1}{x-3} \) is undefined when \( x = 3 \).

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How do you find the values that make a rational expression undefined?

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To find the values that make a rational expression undefined, set the denominator equal to zero and solve for the variable. For example, for \( \frac{1}{x^2 - 4} \), set \( x^2 - 4 = 0 \) to find \( x = 2 \) and \( x = -2 \).

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What is the Greatest Common Factor (GCF)?

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The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. For example, the GCF of 12 and 16 is 4.

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How do you multiply rational expressions?

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To multiply rational expressions, multiply the numerators together and the denominators together. For example, \( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \).

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What is the process for dividing rational expressions?

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To divide rational expressions, multiply by the reciprocal of the second expression. For example, \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \).

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What does it mean to simplify a rational expression?

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Simplifying a rational expression means reducing it to its lowest terms by canceling common factors in the numerator and denominator.

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What is a rational expression?

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A rational expression is a fraction where the numerator and the denominator are polynomials. For example, \( \frac{x^2 + 2x + 1}{x - 1} \) is a rational expression.

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How do you add rational expressions with different denominators?

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To add rational expressions with different denominators, find a common denominator, convert each expression, and then add the numerators. For example, \( \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \).

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What is the difference between a proper and improper rational expression?

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A proper rational expression has a numerator with a lower degree than the denominator, while an improper rational expression has a numerator with a degree equal to or higher than the denominator.

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How do you factor a polynomial?

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To factor a polynomial, look for common factors, use grouping, or apply special factoring formulas like the difference of squares. For example, \( x^2 - 9 = (x - 3)(x + 3) \).