Rational Expressions Student will be able to simplify rational expressions And identify what values make the expression Undefined . a.16

Simplifying Rational Expressions The objective is to be able to simplify a rational expression-These are already!

Undefined denominators Ignore the numerator Set the denominator = to zero and solve

Undefined denominators-ex. What value(s) would make these undefined

Undefined denominators-ex. What value(s) would make these undefined X+2=0 x2 – 9 = 0 X=-2 x+ 3 = 0 x – 3 = 0 x = -3 x = 3

Try these: For what value of a are these undefined:

Answers: 1. 4a = 0 4 4 a = 0 2. 3a+2 = 0 -2 -2 3a = -2 3 3 a = -2/3

This is not reduced: We do not have to factor monomial terms….

The greatest common factor is 5…divide it out both parts….

Try these:

Cancel all common factors…. Answers:

Vocabulary Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are polynomials. Domain of a rational expression – the set of all real numbers except those for which the denominator is zero. Reduced form – a rational expression in which the numerator and denominator have no factors in common.

Simplifying Rational Expressions Divide out the common factors Factor the numerator and denominator and then divide the common factors

Dividing Out Common Factors Step 1 – Identify any factors which are common to both the numerator and the denominator. The numerator and denominator have a common factor. The common factor is the five.

Dividing Out Common Factors Step 2 – Divide out the common factors. The fives can be divided since 5/5 = 1 The x remains in the numerator. The (x-7) remains in the denominator

Factoring the Numerator and Denominator Factor the numerator. Factor the denominator. Divide out the common factors. Write in simplified form.

Factoring Step 1: Look for common factors to both terms in the numerator. 3 is a factor of both 3 and 9. X is a factor of both x2 and x. Step 2: Factor the numerator. 3 x ( x + 3 ) 3 12 x

Factoring Step 3: Look for common factors to the terms in the denominator and factor. The denominator only has one term. The 12 and x3 can be factored. The 12 can be factored into 3 x 4. The x3 can be written as x • x2. 3 x ( x + 3 ) · · · 2 3 4 x x

Divide and Simplify x + 3 4 x 2 Step 4: Divide out the common factors. In this case, the common factors divide to become 1. Step 5: Write in simplified form. x + 3 2 4 x

You Try It Simplify the following rational expressions.

Problem 1 Divide out the common factors. Write in simplified form.

Problem 2 Factor the numerator and denominator You Try It Problem 2 Factor the numerator and denominator Divide out the common factors. Write in simplified form.

Problem 3 Factor the numerator and denominator You Try It Problem 3 Factor the numerator and denominator Divide out the common factors. Write in simplified form.

Problem 4 Factor the numerator and denominator You Try It Problem 4 Factor the numerator and denominator Divide out the common factors. Write in simplified form.

Reducing to -1 Reduce:

Answer: -1

Student will be able to Multiply Rational Expressions and express in simplest form a2.a.16 Do Now: Multiply: Copy this:

Student will be able to Multiply Rational Expressions and express in simplest form a2.a.16 Cross cancel common factors and then multiply across The numerators and across the denominators: 9 4

Multiplying when factoring is necessary!

Canceling step: Cancel top and bottom and on diagonals: 2 Multiply numerators, multiply denominators: =

Ex:

Restrictions on Rational Expressions For what value of x is undefined? It is undefined for any value of “x” which makes the denominator zero. The restriction is that x cannot equal 5.

YOU TRY IT What are the excluded values of the variables for the following rational expressions?

Problem 1 Solution y  0 z  0

Problem 2 Solution 2x - 12 = 0 ANSWER 2x - 12 + 12 = 0 + 12 x  6

More complicated What are the excluded values of the variables for the following rational expression. ? (undefined)

Problem 3 Solution C2 + 2C - 8 = 0 Answer C  2 (C-2)(C+4) = 0 C  -4 C-2 = 0 or C + 4 = 0 C = 2 or C = -4 Answer C  2 C  -4

Dividing Rationals Student will be able to divide rational expressions and Express answer in simplest form. Do Now: divide these fractions (remember that dividing is Multiplying by the reciprocal)

Answer Multiply by the reciprocal: a.k.a.: “Flip” and multiply 2 1

Algebraic Example: Note: after inverting, (“flipping”) the second expression, factor all four parts and follow multiplying rules

Algebraic Example: 2

Example 2 (Completely factor the First numerator)

Example 2 (Completely factor the First numerator) 2

Do and hand in on exit card:

Adding/Subtracting Rational Expressions Do now: (remember common denominators) Today, you will be able to add rational expressions by finding Least common denominators…..

Adding/Subtracting Rational Expressions

Algebraic examples:

Algebraic examples: Lcd = 6 Lcd=6

Answers: Distribute!

Subtracting-remember to distribute!

Subtracting-remember to distribute! But this can be reduced!

Reducing:

Trickier denominators: Here we should factor the second denominator in order to find The least common denominator…

Finding the lcd: Which means (x+3)(x – 3) is the lcd so multiply the first Fraction by (x – 3)/(x – 3)

Answer: Not reducable!

Next example:

Solution:

Try this-(factor to find lcd) This one will need to be reduced at the end….

Answer lcd = (x-5)(x+3):

Answer:

Complex Fractions a.17 Student will be able to simplify complex fractions by Multiplying each term by the least common denominator and Simpifying if necessary. Do Now - Divide:

A fraction over another fraction Now think of it this way: This is called a complex fraction. We flip the bottom and multiply, just Like when we divided.

Fractions within a fraction: Step 1-find the lcd of all 4 terms Step 2-multiply each term by the lcd/1

Fractions within a fraction: Step 1-find the lcd of all 4 terms Step 2-multiply each term by the lcd/1 x 1 Lcd – x2 1 x

Example:

Solution: lcd = b2

Solution: lcd = b2 b b

Next example:

lcd: ab But this one needs to be reduced!

lcd: ab

Solving Rational equations: Do now: page 60 # 11,12

Solving rational equations using the lcd method: How is this different than the ones you just solved? Find the lcd of all terms Multiply each term by the lcd Solve the equation STEPS:

Solution: 2 3 6 + 2a = 9 – 6a Look, we eliminated denominators!

+6a +6a -6 -6 ____ ___ 8 8

Example:

Lcd=2x 4x + 6 = 10 4x = 4 x = 1

Try this:

Lcd=a(a+2) a2=3a+6+4 a2=3a+10 a2-3a-10=0 (a-5)(a+2)=0 a=5, a=-2

Extraneous roots: Sometimes, when we check roots in the original Equation, we arrive at an undefined denominator. These are called extraneous roots. Check the roots in the previous problem Which one is extraneous? Why?

Review Students will review rational expressions and equations Do Now: Solve for x:

Review Rationals-index card review problems Multiply and express in simplest form: 1. For what value of x is this undefined? 2.

Review Rationals Add or subtract and express in simplest form: 3. 4. 5. Express this complex fraction in simplest form:

Solving: 6.

Answers Add corrected problems to index card for folder…

Finding the LCD It is sometimes necessary to factor the denominators!