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Rational Expressions

This comprehensive guide covers rational functions, their properties, and graphical behavior. Learn how to identify vertical and horizontal asymptotes, simplify rational expressions, multiply and divide them, and find the least common denominator (LCD). Discover the concept of complex fractions and how to resolve them. The text also provides step-by-step procedures for solving rational equations, ensuring that you check each solution for accuracy. Perfect for mastering the fundamentals of rational expressions in algebra.

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Rational Expressions

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  1. RationalExpressions

  2. Rational functions and their graphs • A rational function can be written as f(x)=where P(x) and Q(x) are polynomial functions and Q(x) can not equal zero. • Example:

  3. Vertical and horizontal asymptopes • To find vertical asymptotes, set the denominator equal to zero. • A graph ahs at most one horizontal asymptote if: (the m degree of numerator and n is degree of denominator) • m<n, then y=0 is the horizontal asymptote • m=n, then line y= • m>n, then there is no horizontal asymptote

  4. Rational expressions • A rational expression is in simplest form when its numerator and denominator are polynomials that have no common divisors. • To get an expression in simplest form, divide out the common factors. • Ex: • Factor: • Divide out common factors to get the answer • Set the common factors equal to zero to get restrictions. Restrictions are at x=-4 or -5

  5. Multiplying • When multiplying, use what you know about simplifying rational expressions to work them. Multiply across. • Ex: x • Ex: X

  6. Dividing rationals When dividing, use the term “copy, switch, flip”. Copy the first term of the expression, switch the dividing sign to a multiplication sign, and flip the last expression to its reciprocal. • Ex: / • Ex: /

  7. Finding least common multiples • To add or subtract, you have to find the LCD. To do this, find the least common multiple. • Ex: Find the least common multiple of each pair of polynomials. and . (x+1)(x-1) (x+1)(x+1) Both pairs have a (x+1) in common. To find LCD, multiply each by what they’re missing. Multiply the first pair by the other (x+1) that it’s missing, and multiply the second pair by the (x-1) that it’s missing. The LCD would end up being (x+1)(x+1)(x-1).

  8. Adding and subtracting expressions • Find the LCD of the expressions to add and subtract. Add across the numerators and simplify. • Ex: Factor the denominators and find the LCD Multiply numerator and denominator by what each expression is missing in the LCD

  9. Adding and subtracting ex: • Ex: + • Ex: • Ex: • Ex: -

  10. Complex fractions • A Complex Fraction is a fraction that has a fraction in its numerator or denominator. Its can also be found in both numerator and denominator. • Examples: • To simplify a complex fraction such as you can multiply the numerator and denominator by their LCD bd. Or you can divide the numerator by the denominator .

  11. Simplifying complex fractions Ex: • First find the LCD of all the rational expressions. = • The LCD is xy. Multiply the numerator and denominator by xy. = • Use the distributive property = • Simplify

  12. Try it yourself…

  13. Solving rational equations • Solve and then check each solution. = Write cross products… 5(x^2-1) = 5(2x-2) Distributive Property… 5x^2-5 = 30x – 30 Write in standard form…5x^2 – 30x +25 = 0 Divide each side by 5… x^2 – 6x +5 = 0 Factor… (x-1)(x-5) = 0 Zero-Product Property… x=1 or x=5

  14. Try it yourself… = =

  15. Unit 3Be ready to go when the bell rings! Example 1:

  16. Unit 3 Example 2:

  17. Unit 3 Example 3:

  18. Unit 3 Example 4:

  19. Unit 3 Example 5:

  20. Unit 3 Example 6: xint: VA: HA: Hole:

  21. Unit 3 Example 7: Describe the translation, list both asymptotes.

  22. MSL (Common Exam)

  23. MSL (Common Exam)

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