Do Now Factor completely. 1. 2. 3. 4. 18𝑥𝑦( 3𝑥 3 − 𝑦 3 )

Assignment: P.6 Rational Expressions Day 2 8/23/16 P.6 Rational Expressions Objectives: Simplify rational expressions and operations on rational expressions. Assignment:

Rational Expressions Rational Expressions A rational expression is the quotient of two polynomials. The set of real numbers for which an algebraic expression is defined is the domain of the expression. Because rational expressions indicate division and division by zero is undefined, we must exclude numbers from a rational expression’s domain that make the denominator zero. DENOMINATOR can not = 0

Example: Excluding Numbers from the Domain Rational Expressions Find all the numbers that must be excluded from the domain of the rational expression: To determine the numbers that must be excluded from the domain, examine the denominator. We set each factor equal to zero to find the excluded values.

You DO Find all the numbers that must be excluded from the domain. 1) 4 𝑥−2 2. 𝑥 𝑥 2 −1 3. 𝑥 𝑥 2 +11𝑥+10 (x-1)(x+1) (x+10)(x+1) 1. x≠ 2 2. x ≠ -1, x ≠1 3. x ≠-10, x ≠-1

Example: Simplifying Rational Expressions Check for excluded values. Because the denominator is

𝑥 3 + 𝑥 2 𝑥+1 = 𝑥 2 , x≠-1 3. 𝑥 2 −1 𝑥 2 −2𝑥+1 2. 𝑥 2 +6𝑥+5 𝑥 2 −25 𝑥 3 + 𝑥 2 𝑥+1 = 𝑥 2 (𝑥+1) (𝑥+1) You DO 1. = 𝑥 2 , x≠-1 2. 𝑥 2 +6𝑥+5 𝑥 2 −25 = (𝑥+5)(𝑥+1) (𝑥−5)(𝑥+5) = 𝑥+1 𝑥−5 , x ≠ 5, x ≠ -5 3. 𝑥 2 −1 𝑥 2 −2𝑥+1 = (𝑥+1)(𝑥−1) (𝑥−1)(𝑥−1) = 𝑥+1 𝑥−1 , 𝑥 ≠ 1

Multiplying Rational Expressions 1. Factor all numerators and denominators completely. 2. Divide numerators and denominators by common factors. 3. Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators.

Example: Multiplying Rational Expressions 𝑥+3 𝑥 2 −4 ∗ 𝑥 2 −𝑥−6 𝑥 2 +6𝑥+9 Multiply: Factor the numerator and the denominator: Check for excluded values. Because the denominator is ∗ ∗ Divide numerators and denominators by common factors: ∗ ∗

= (𝑥−3)(𝑥+3) 𝑥 2 ∗ 𝑥(𝑥−3) (𝑥+4)(𝑥−3) 𝑥 2 −9 𝑥 2 ∗ 𝑥 2 −3𝑥 𝑥 2 +𝑥−12 You Do Factor everything Exclude values of x Cancel 1. = (𝑥−3)(𝑥+3) 𝑥 2 ∗ 𝑥(𝑥−3) (𝑥+4)(𝑥−3) 𝑥 2 −9 𝑥 2 ∗ 𝑥 2 −3𝑥 𝑥 2 +𝑥−12 1 (𝑥−3)(𝑥+3) 𝑥(𝑥+4) ,X≠-4,0, 3

= ( 𝑥 2 +2𝑥+4) 3𝑥 𝑥 3 −8 𝑥 2 −4 ∗ 𝑥+2 3𝑥 𝑥≠−2, 0, 2 ,𝑥≠−2, 0, 2 𝑥 3 −8 𝑥 2 −4 ∗ 𝑥+2 3𝑥 = 𝑥 −2 ( 𝑥 2 +2𝑥+4) 𝑥−2 ( 𝑥+2) ∗ (𝑥+2) 3𝑥 𝑥≠−2, 0, 2 = 𝑥 −2 ( 𝑥 2 +2𝑥+4) 𝑥−2 ( 𝑥+2) ∗ (𝑥+2) 3𝑥 = ( 𝑥 2 +2𝑥+4) 3𝑥 ,𝑥≠−2, 0, 2

Dividing Rational Expressions The quotient of two rational expressions is the product of the first expression and the multiplicative inverse, or reciprocal, of the second expression. The reciprocal is found by interchanging the numerator and the denominator. Thus, we find the quotient of two rational expressions by inverting the divisor and multiplying. Keep change Flip

Example: Dividing Rational Expressions Divide: Invert the divisor and multiply. Factor as many numerators and denominators as possible. ● ● ● Because the denominator is

Example: Dividing Rational Expressions (continued) Divide: Divide numerators and denominators by common factors. = 3(𝑥−1 𝑥(𝑥+2 ,𝑥≠0,𝑥≠1,𝑥≠−2

𝑥+2 𝑥−3 , 𝑥≠−3, 𝑥≠3, 𝑥≠4 1. 𝑥+3 𝑥 2 −4 ∗ 𝑥 2 −𝑥−6 𝑥 2 +6𝑥+9 Divide or Multiply Change this problem 1. 𝑥+3 𝑥 2 −4 ∗ 𝑥 2 −𝑥−6 𝑥 2 +6𝑥+9 2. 𝑥 2 −2𝑥−8 𝑥 2 −9 ÷ 𝑥−4 𝑥+3 𝑥+2 𝑥−3 , 𝑥≠−3, 𝑥≠3, 𝑥≠4

3. 𝑥 2 +𝑥−12 𝑥 2 +𝑥 −30 ∗ 𝑥 2 +5𝑥+6 𝑥 2 −2𝑥 −3 ÷ 𝑥+3 𝑥 2 +7𝑥+6 3. 𝑥 2 +𝑥−12 𝑥 2 +𝑥 −30 ∗ 𝑥 2 +5𝑥+6 𝑥 2 −2𝑥 −3 ÷ 𝑥+3 𝑥 2 +7𝑥+6 4. , x≠ -5,-2,-1, 1

3𝑥 2 +4𝑥+1 3𝑥 2 −5𝑥−2 ÷ 𝑥 2 −2𝑥−3 −5𝑥 2 +25𝑥−30

Adding and Subtracting Rational Expressions with the Same Denominator Adding and Subtracting Rational Expressions with Like Denominators To add or subtract rational expressions with the same denominator: 1. Add or subtract the numerators. 2. Place this result over the common denominator. 3. Simplify, if possible. Reminder: check for excluded values.

Example: Subtracting Rational Expressions with the Same Denominator Adding and Subtracting Rational Expressions with Like Denominators Subtract: Check for excluded values:

Adding and Subtracting Rational Expressions with Unlike Denominators Adding and Subtracting Rational Expressions That Have Different Denominators Adding and Subtracting Rational Expressions with Unlike Denominators 1. Find the LCD of the rational expressions. 2. Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD. 3. Add or subtract numerators, placing the resulting expression over the LCD. 4. If possible, simplify the resulting rational expression.

Example: Subtracting Rational Expressions with Different Denominators Adding and Subtracting Rational Expressions with Unlike Denominators Subtract: Step 1 Find the least common denominator. The LCD is Step 2 Write equivalent expressions with the LCD as denominators.

Example: Subtracting Rational Expressions with Different Denominators Adding and Subtracting Rational Expressions with Unlike Denominators Subtract: Step 3 Subtract numerators, putting this difference over the LCD. Step 4 If necessary, simplify.

You Do 1. 2. 𝑥−3 𝑥 2 −1 − 𝑥+2 𝑥−1

= 3𝑥 (𝑥+5)(𝑥−2) - 2𝑥 (𝑥+3)(𝑥−2) = You Do Find the LCD- exclude values for x Make equivalent fractions using the LCD Add or Subtract Simplify 3𝑥 𝑥 2 +3𝑥−10 − 2𝑥 𝑥 2 +𝑥−6 = 3𝑥 (𝑥+5)(𝑥−2) - 2𝑥 (𝑥+3)(𝑥−2) = 3𝑥(𝑥+3) (𝑥+5)(𝑥−2)(𝑥+3) − 2𝑥(𝑥+5) (𝑥+3)(𝑥−2)(𝑥+5) = 3𝑥 𝑥+3 −[2𝑥 𝑥+5 ] (𝑥+5)(𝑥−2)(𝑥+3) = 3𝑥 2 +9𝑥− 2𝑥 2 −10𝑥 (𝑥+5)(𝑥−2)(𝑥+3) = 𝑥 2 −𝑥 (𝑥+5)(𝑥−2)(𝑥+3) 𝑥 ≠ -5,-3,2

Do Now

Complex Rational Expressions A complex fraction or complex rational expression is one that has AT LEAST one fraction in the NUMERATOR or Denominator or IN BOTH the numerator and denominator. Below are some examples.

There are two methods to simplify complex fractions 𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2

𝑥 2 +1 𝑥 ÷ 𝑥+3 𝑥 2 = 𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2 = 𝑥 2 +1 𝑥 𝑥+3 𝑥 2 Method 1 Simplify the numerator and the denominator 𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2 = 𝑥 2 +1 𝑥 𝑥+3 𝑥 2 Rewrite the problem 𝑥 2 +1 𝑥 ÷ 𝑥+3 𝑥 2 =

𝑥 2 +1 𝑥 ÷ 𝑥+3 𝑥 2 = 𝑥 2 +1 𝑥 ∗ 𝑥 2 𝑥+3 = 𝑥 2 𝑥 2 +1 𝑥 𝑥+3 Rewrite the problem 𝑥 2 +1 𝑥 ÷ 𝑥+3 𝑥 2 = 𝑥 2 +1 𝑥 ∗ 𝑥 2 𝑥+3 = 𝑥 2 𝑥 2 +1 𝑥 𝑥+3 = 𝑥 𝑥 2 +1 (𝑥+3)

𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2 𝑥+ 1 𝑥 ∗ 𝑥 2 ( 1 𝑥 + 3 𝑥 2 )∗ 𝑥 2 = 𝑥 3 +𝑥 𝑥+3 Method 2 Find the LCD & MULTIPLY the numerator and denominator by the LCD 𝑥+ 1 𝑥 1 𝑥 + 3 𝑥 2 𝑥+ 1 𝑥 ∗ 𝑥 2 ( 1 𝑥 + 3 𝑥 2 )∗ 𝑥 2 = 𝑥 3 +𝑥 𝑥+3 = 𝑥 𝑥 2 +1 𝑥+3

Example: Simplifying a Complex Rational Expression Simplifying Rational Expressions Simplify:

2 3 1 2 1 𝑥 + 1 𝑦 𝑥+𝑦 Divide: Your Turn 1+ 1 𝑥 1− 1 𝑥 1+ 1 𝑥 1− 1 𝑥 = 𝑥 𝑥 + 1 𝑥 𝑥 𝑥 − 1 𝑥 = 𝑥+1 𝑥 𝑥−1 𝑥 1) 2 3 1 2 2) 1 𝑥 + 1 𝑦 𝑥+𝑦 3)

4.

5. 6. 3 𝑥−2 − 4 𝑥+2 7 𝑥 2 −4 1 𝑥 − 3 2 1 𝑥 + 3 4 7. 1 (𝑥+ℎ) 2 − 1 𝑥 2 ℎ

Fractional Expressions in Calculus Fractional expressions containing radicals occur frequently in calculus. These expressions can often be simplified using the procedure for simplifying complex rational expressions Simplify:

Your Turn: 9−𝑥 2 + 𝑥 2 9−𝑥 2 9 −𝑥 2

Rationalizing the Numerator To rationalize a numerator, multiply the numerator and the denominator by the conjugate of the numerator. Rationalize the numerator:

𝑥+ℎ − 𝑥 ℎ h ≠0 but what happens to the expression as h approaches 0