Unit #4 Rational Expressions Chapter 5 Sections 2-5

Warm Up Simplify each expression. Assume all variables are nonzero. 1. x5  x2 x7 2. y3  y3 y6 x6 x2 y2 y5 1 y3 3. x4 4. Factor each expression. 5. x2 – 2x – 8 (x – 4)(x + 2) 6. x2 – 5x x(x – 5) 7. x5 – 9x3 x3(x – 3)(x + 3)

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!!! Examples>

Simplify Rational Expressions To Simplify: Apply the same rules as simplifying fractions First step is identify & eliminate common factors Ex>

Simplifying Rational Expressions Examples: Factor all expressions (numerator/denominator) before crossing out common factors. Ex> Common Mistakes *Be Careful*

Factoring out (-1) to cross our factors Sometimes factors look like they can cross out BUT need some signs to change. Evaluate: True or False Ex>

Multiplying Formula: Set up with numerators above the line and denominators below the line Expression w/o denominator is over 1 Completely FACTOR numerators & denominators Divide out (cross out) common factors (top and bottom) Ensure INITIAL expression does not have 0 on bottom Ex>

Dividing Formula: Assume all rational expressions are defined Rewrite expression - multiply by the reciprocal KEEP CHANGE FLIP First Expression Divide to Mult 2nd Expression Then factor and simplify like multiplication Ex>

Solving Simple Rational Expressions Factor/Simplify/Solve Example:

Warm Up Add or subtract. 7 15 2 5 + 13 15 1. 11 12 3 8 – 2. 13 24 Simplify. Identify any x-values for which the expression is undefined. 4x9 12x3 3. 1 3 x6 x ≠ 0 1 x + 1 x – 1 x2 – 1 4. x ≠ –1, x ≠ 1

Adding/Subtracting Rational Expressions (5.3) Adding and subtracting rational expressions is similar to adding and subtracting fractions. To add or subtract rational expressions with like denominators, add or subtract the numerators and use the same denominator.

Like Denominators Ex> Ex> Ex> Ex> *Remember to Simply*

Adding/Subtracting Rational Expressions (5.3) To add or subtract rational expressions with unlike denominators, treated same as ordinary fractions. Must have a common denominator (LCD) LCD is composed of the least common multiple (LCM) What does one side have that the other side needs. Completely factor all denominators LCM is the combination of all factors – use highest power of any common factor

Least Common Multiples (5.3) Least Common Multiple has to contain ALL the factors of both expressions Use all of the different factors if polynomials have common factors use the HIGHEST POWER of each common factor Monomials – 6x2y3 & 4x4y2z Polynomials – y+3 & y2+2y-3 Polynomials – x2+3x-4 & x2-3x+2

Without Common Denominators Find common denominator (LCM) Multiply top & bottom of term by missing factor What does one side have that the other side needs. What you do to the denominator of a fraction you MUST DO to the numerator Add or subtract numerators with common denominator Combine like terms – factor & simplify Ex>

Without Common Denominators cont. Ex>

Add & Subtract Example Subtract . Identify any x-values for which the expression is undefined. 2x2 + 64 x2 – 64 – x + 8 x – 4 2x2 + 64 (x – 8)(x + 8) – x + 8 x – 4 Factor the denominators. The LCD is (x – 3)(x + 8), so multiply by . x – 4 x + 8 (x – 8) 2x2 + 64 (x – 8)(x + 8) – x + 8 x – 4 x – 8 2x2 + 64 – (x – 4)(x – 8) (x – 8)(x + 8) Subtract the numerators. Multiply the binomials in the numerator. 2x2 + 64 – (x2 – 12x + 32) (x – 8)(x + 8)

Example Continued Subtract . Identify any x-values for which the expression is undefined. 2x2 + 64 x2 – 64 – x + 8 x – 4 2x2 + 64 – x2 + 12x – 32) (x – 8)(x + 8) Distribute the negative sign. x2 + 12x + 32 (x – 8)(x + 8) Write the numerator in standard form. (x + 8)(x + 4) (x – 8)(x + 8) Factor the numerator. x + 4 x – 8 Divide out common factors. The expression is undefined at x = 8 and x = –8 because these values of x make the factors (x + 8) and (x – 8) equal 0.

Complex Fractions One or more fractions in the numerator, the denominator or both The bar represents division so one way to solve is to rewrite as division then simplify. Also simplify by multiplying top & bottom by LCD of fractions in numerator & denominator

Example: Simplifying Complex Fractions Simplify. Assume that all expressions are defined. x + 2 x – 1 x – 3 x + 5 Write the complex fraction as division. x + 2 x – 1 x – 3 x + 5 ÷ Write as division. Multiply by the reciprocal. x + 2 x – 1 x + 5 x – 3  x2 + 7x + 10 x2 – 4x + 3 (x + 2)(x + 5) (x – 1)(x – 3) or Multiply (FOIL).

Example 2: Simplifying Complex Fractions Simplify. Assume that all expressions are defined. x – 1 x 2 3 + Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator. x – 1 x (2x) 2 3 + The LCD is 2x.

Example with speed or work Example 2 Continued Simplify. Assume that all expressions are defined. (3)(2) + (x)(x) (x – 1)(2) Divide out common factors. x2 + 6 2(x – 1) x2 + 6 2x – 2 or Simplify. Example with speed or work

Solving Rational Expressions (5.5) One term Rational Expressions on both sides Cross-multiply and solve using algebra Remember to always check for extraneous solutions (solutions that would make denominator equal to zero) Ex>

Solving Rational Expressions (5.5) Equation with more than one term on each side Remember to always check for extraneous solutions (solutions that would make denominator equal to zero) 2 step process 1) Mult. each piece of the equation by the LCM (numerator only). This will eliminate all denominators 2) Solve using Algebra Ex>

Solving Rational Expressions (5.5) Extraneous Solutions Once you solve for the value of x, ensure the answer DOES NOT RESULT IN 0 IN DENOMINATOR Ex>

Review And/Or Statements When solving rational inequalities you must remember the your answers will be a range of answers. These answer are written as and/or statements. Just like quadratic inequalities. AND OR Be Careful because the extraneous solutions must not be an equals to inequality.

Rational Inequalities (5.5) *Range of Answers* Solving Using Graph and Table (Calculator) Plug left side into Y1 Plug right side into Y2 Look at graph and table for where condition is met Ex> Ex>

Rational Inequalities (5.5) *Range of Answers* Solving using Algebra (5 steps) 1)Determine excluded values (denominator = 0) 2)Set the inequality into an equation 3)Solve the equation 4)Put all numbers on a number line (excluded & solutions) 5)Check regions for when condition is met Ex>

Asymptotes & Transformations X Y -4 4 -3 3 -2 2 -1 1 -.5 .5 -.25 .25 -.125 .125 Parent Function : Vertical Asymptote- vertical line (x=#) that the function approaches (VA) Horizontal Asymptote – horizontal line (y=#) that the function approaches (HA)

Transformations Domain: {x I x ≠ h} Range: {y I y ≠ k} 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} Name the Transformation, Identify VA & HA, Identify D: & R: Ex> Ex> Ex>

Zeros, and Asymptotes If the Rational Expression is in the form: p(x) = where you find your zeros (set numerator equal to zero and solve) q(x) = where you find your VA (set denominator equal to zero and solve) *x=* Horizontal Asymptote Degree of top compared to Degree of bottom Numerator < Denominator – x axis (y=0) Numerator = Denominator – LC Top/LC Bottom (y=#) Numerator > Denominator – NO Horizontal Asymptote

Zeros, Holes, and Asymptotes Look at degree of numerator and denominator and determine HA. (use the chart) Completely Factor the Rational Function If there is a common factor cross them out and set the common factor to zero. (Holes) Find Zeros – Set numerator to zero and solve Find VA – Set denominator to zero and solve Ex> Ex> Ex>