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Aim 2-1: How do we define and simplify rational expressions? HWk read 2-1 p 67# 1-10 Objective: SWBAT Simplify a Rational Expression.

Objectives: 1.SWBAT to define a rational expression. 2.SWBAT to simplify a rational expression to its simplest form. 3. SWBAT to find prohibited values for variables in denominators that would make an expression undefined. 4. SWBAT to +, -, * and ÷ with rational expressions and then to factor and to simplify rational expressions.

Review: Laws of Exponents Multiplying Powers: *

Laws of Exponents Dividing Powers: Power of a Power: Power of a Product:

Laws of Exponents Negative Exponents:

Laws of Exponents Power of a Quotient: Power of Zero: 1

Examples using the laws of exponents:

Definition: A Rational Expression is the quotient of two polynomials with the denominator not equal to zero.

EXCLUDED VALUES IN DENOMINATORS Any value of x that makes the denominator = 0 is prohibited from the expression. Why? Because an expression is undefined when its denominator is equal to 0.

For test show algebra:x+ 2 = x = - 2

If x= -1, then = 0. The expression is undefined when a variable value makes the denominator equal to 0

RECALL: Rational number = any number that may be expressed as a quotient of two integers with no 0 denominator. Now we have: Rational expression = any expression that may be stated as a quotient of two polynomials with no 0 denominator.

Remember, denominators cannot = 0. Now, lets go through the steps to simplify a rational expression.

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify.

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Would you like to review factoring of trinomials?

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Are the polynomials in ax 2 + bx + c form?

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Yes, each polynomial is in ax 2 + bx + c form? So for x 2 + 6x + 5 we need: ___ + ___ = b ___ * ___ = c

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Yes, each polynomial is in ax 2 + bx + c form? So for x 2 + 6x + 5 we need: ___ + ___ = 6 ___ * ___ = 5

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Yes, each polynomial is in ax 2 + bx + c form? So for x 2 + 6x + 5 we need: _5__ + _1__ = 6 _5__ * _1__ = 5 so our numerator is (x+5)(x+1) x

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Can we factor the denominator? Do you recognize DOTS? x 2 + 6x + 5 = (x+5)(x+1) x x

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Can we factor the denominator? Do you recognize DOTS? a 2 – b 2 = (a-b)(a+b) x 2 + 6x + 5 = (x+5)(x+1) x x

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Can we factor the denominator? Do you recognize DOTS? a 2 – b 2 = (a-b)(a+b) So our denominator of x 2 – 25 = (x - )(x + ) x 2 + 6x + 5 = (x+5)(x+1) x x

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Can we factor the denominator? Do you recognize DOTS? a 2 – b 2 = (a-b)(a+b) So our denominator of x 2 – 25 = (x -5)(x +5 ) x 2 + 6x + 5 =(x+5)(x+1) = (x+5)(x+1) x x 2 – 25 (x-5)(x+5)

Writing a rational expression in simplest form. Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify. Can we cancel like binomials as like factors? Yes! Our final answer is = x+1 x-5 x 2 + 6x + 5 =(x+5)(x+1) = (x+5)(x+1) x x 2 – 25 (x-5)(x+5)

Step 1: Factor the numerator and the denominator completely looking for common factors. Next

What is the common factor? Step 2: Divide the numerator and denominator by the common factor.

1 1 Step 3: Cancel and simplify.

How do I find the values that make an expression undefined? Completely factor the original denominator.

How do we determine when this is undefined? Cross out Numerator. Factor the denominator

Set factors = 0 one at a time and solve. The expression is undefined when: a= 0, 2, and -2 and b= 0. End test #2. Factor the denominator

On the Regents EXAM, “Simplest form” means all common factors have been canceled. So, Step 1: Factor both numerator and denominator completely. Step 2: Cancel common factors and simplify.

Lets go through another example. Put this expression in simplest form. Factor out the GCF Next

1 1 KEY TRICK

For what values will the original expression be undefined? Go back to prior slide and set factors = 0.

Now try to do some on your own. Put these in their simplest form. Also find the values that make each expression undefined? Time permitting start hwk.

Multiplying Rational Expressions. With rational expressions, we always factor first and then cancel common factors in numerators and denominators before we multiply

Let’s do another one. Step #1: Factor the numerator and the denominator. Next

Step #2: Divide the numerator and denominator by the common factors

Step #3: Multiply the numerator and the denominator. Next: division of rational expressions.

Recall how to divide by a fraction: Multiply by the reciprocal of the divisor AKA: Keep Flip Change

Flip divisor and rewrite the problem as multiplication. Factor each rational expression Divide out the common factors. Write in simplified form. Division of rational expressions.

Flip divisor and rewrite the problem as multiplication. Factor each rational expression Divide out the common factors. Write in simplified form. Division of rational expressions.

Factor each rational expression Divide out the common factors. Write in simplified form. Flip divisor and rewrite the problem as multiplication. Division of rational expressions.

Factor each rational expression Divide out the common factors. Write in simplified form. Flip divisor and rewrite the problem as multiplication. Division of rational expressions. = 1

Factor each rational expression Divide out the common factors. Write in simplified form. Division of rational expressions. Flip divisor and rewrite the problem as multiplication.

Factor each rational expression Divide out the common factors. Write in simplified form. Division of rational expressions. Flip divisor and rewrite the problem as multiplication.

Ex: Simplify

Next Keep-Flip-Change

Now you try to simplify the expression: Keep-Flip-Change

Now try these on your own. Keep-Flip-Change

Here are the answers: