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Understanding Rational Expressions and Domain Restrictions

Rational expressions are formed by the ratio of two polynomials, with the denominator not equal to zero. The domain of a rational expression includes all real numbers except for those values that make the denominator zero. For example, if you encounter a rational expression defined as ( frac{p(x)}{q(x)} ), the domain excludes any values of ( x ) for which ( q(x) = 0 ). Understanding the domain is essential for determining valid inputs for your algebraic expression and ensuring that calculations are accurate and defined.

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Understanding Rational Expressions and Domain Restrictions

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  1. Rational Expressions Section 0.5

  2. Rational Expressions and domain restrictions • Rational number- ratio of two integers with the denominator not equal to zero. • Rational expression- ratio or quotient of two polynomials with the denominator not equal to zero Examples: Rational number: Rational expression: where x = 6

  3. Domain- set of real numbers that your algebraic expression is defined. • Think about domain as what values are OK to plug into your equation. • For rational expressions our domain will not be defined for the values that make the denominator zero. • What is the domain for: Answer: All Real numbers except x = -3

  4. Find the domain for each algebraic expression Domain: All real numbers Domain: All real numbers except x = 0 Domain: All real numbers except Domain: All real numbers

  5. Find the domain for each algebraic expression Domain: All real numbers except x = 0 and x = 5 Domain: All real numbers except x = 4 and x = -4

  6. Reduce the rational expression Where x = -5 Where x = -1

  7. Multiply the rational expressions and simplify Check domain at factored step: Domain: All reals except:

  8. Multiply Domain Restrictions:

  9. Divide the rational expressions Domain: All reals except -2, 0, and 2

  10. Divide Domain: All reals except 0, 3 and -3

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