Rational Expressions

Rational Expressions (5.2) QUOTIENT of 2 polynomial expressions – Ratio (or Fraction) of 2 polynomials – Function with variables (x) on the bottom A Rational Expression cannot have a ZERO on the bottom!!! To Simplify: – First step is identify & eliminate common factors – x/(x+2) or (x 2 +2x-3)/(x 2 +6x-9)

Simplifying Rational Expressions Examples: Monomial 4xy Monomial 2x 2 Monomial 4xy Polynomial 2x – 3 Polynomial x Polynomial x 2 -3x+2 Polynomial 2x - x 2 Polynomial x 2 -x-2

Multiplying Set up with numerators above the line and denominators below the line – Expression w/o denominator is over 1 Factor numerators & denominators completely Divide out common factors (top and bottom) Ensure INITIAL expression does not have 0 on bottom Examples: 2x 2 y 3 * 9x 4 x + 2 * x + 4 3x 5 4y 5 3x+12 x 2 – 4

Dividing Assume all rational expressions are defined Rewrite expression - multiply by the reciprocal KEEP CHANGE FLIP First Expression Mult to Divide 2 nd Expression Then factor and simplify like multiplication Examples: 2x 2 y 3 ÷ 4y 4 m 2 -2m-8 ÷ 2m-8 3x 5 9x 5 8m + 24 m 2 +7m+12

Solving Simple Rational Expressions Factor Simplify Solve Example: X 2 – 3x – = 8 x - 5

Adding/Subtracting Rational Expressions (5.3) With Common Denominator – Add the numerators and combine like terms Example:

Least Common Multiples (5.3) Least Common Multiple has to contain ALL the factors of both expressions Monomials – 6x 2 y 3 & 4x 4 y 2 z Polynomials – y+3 & y 2 +2x-3

Adding/Subtracting Rational Expressions (5.3) Without Common Denominator – Find common denominator (LCM) – Multiply top & bottom of term by missing factor – Add or subtract with common denominator – Combine like terms – factor & simplify as able Example:

Asymptotes & Transformations Parent Function : f(x) = 1/x XYXY

Transformations Rewrite as a – f(x) = k (x-h) a: Vertical Stretch/Compression h: Vertical Asymptote Left (x+h) or Right (x-h) k: Horizontal Asymptote Up (+k) or Down (-k) Domain: {x I x ≠ h} Range: {y I y ≠ k}

Asymptotes & Holes Horizontal Asymptote – Degree of top compared to bottom Numerator < Denominator – x axis Numerator = Denominator – LC Top/LC Bottom Numerator > Denominator – NO Horizontal Asymptote Factor Top & Bottom – Roots (Zeros) Factor just on the Top – Vertical Asymptote Factor just on the Bottom – Holes Common Factor on Top & Bottom

Solving Rational Expressions Monomials on both sides – Cross-multiply and solve using algebra – Example: Binomial with Rational Expression – Find Lowest Common Denominator (LCD) – Multiply both sides (all terms) by LCD – Solve using algebra – Example:

Solving Rational Expressions Extraneous Solutions – Once you solve for the value of x, ensure the answer DOES NOT RESULT IN 0 IN DENOMINATOR – Example:

Rational Inequalities Solving Using Graph and Table (Calculator) – Plug Rational Expression side into Y 1 – Plug answer into Y 2 – Look at graph and table for where condition is met – Example:

Rational Inequalities Solving using Algebra – Determine excluded values (denominator = 0) – Set inequality into equation – Solve equation – Put both numbers on number line (excluded & solutions) – Check regions for when condition is met – Example: