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Simplifying. Rational Expressions. Multiplying and Dividing Rational Expressions. Remember that a rational number can be expressed as a quotient of two integers. A rational expression can be expressed as a quotient of two polynomials. Simplifying Rational Expressions.
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Simplifying Rational Expressions
Multiplying and Dividing Rational Expressions Remember that a rational number can be expressed as a quotient of two integers. A rational expression can be expressed as a quotient of two polynomials.
Simplifying Rational Expressions • A “rational expression” is the quotient of two polynomials. (division)
Simplifying Rational Expressions • A “rational expression” is the quotient of two polynomials. (division) • A rational expression is in simplest form when the numerator and denominator have no common factors (other than 1)
Simplifying Rational Expressions • A “rational expression” is the quotient of two polynomials. (division) • A rational expression is in simplest Form when the numerator and denominator have no common factors (other than 1)
How to get a rational expression in simplest form… • Factor the numerator completely (factor out a common factor, difference of 2 squares, bottoms up) • Factor the denominator completely (factor out a common factor, difference of 2 squares, bottoms up) • Cancel out any common factors (not addends)
Difference between a factor and an addend • A factor is in between a multiplication sign • An addend is in between an addition or subtraction sign Example: x + 33x + 9 x – 9 6x + 3
Remember, denominators can not = 0. Now,lets go through the steps to simplify a rational expression.
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: Multiply to get your answer.
Looking at the answer from the previous example, what value of x would make the denominator 0? x= -1 The expression is undefined when the values make the denominator equal to 0
How do I find the values that make an expression undefined? Completely factor the original denominator.
Factor the denominator The expression is undefined when: a= 0, 2, and -2 and b= 0.
Lets go through another example. Factor out the GCF Next
1 1
Now try to do some on your own. Also find the values that make each expression undefined?
Remember how to multiply fractions: First you multiply the numerators then multiply the denominators.
1 1 1 1 1 1 1 1 1 The same method can be used to multiply rational expressions.
Let’s do another one. Step #1: Factor the numerator and the denominator. Next
1 1 1 1 1 1 Step #2: Divide the numerator and denominator by the common factors.
Step #3: Multiply the numerator and the denominator. Remember how to divide fractions?
1 5 1 4 Multiply by the reciprocal of the divisor.
Dividing rational expressions uses the same procedure. Ex: Simplify
1 1 1 1 Next
ObjectivesThe student will be able to: 1. simplify square roots, and 2. simplify radical expressions. The blank boxes are missing * (multiplication signs)
If x2 = y then x is a square rootof y. In the expression , is the radical signand64 is the radicand. 1. Find the square root: 8 2. Find the square root: -0.2
3. Find the square root: 11, -11 4. Find the square root: 21 5. Find the square root:
6. Use a calculator to find each square root. Round the decimal answer to the nearest hundredth. 6.82, -6.82
What numbers are perfect squares? 1 • 1 = 1 2 • 2 = 4 3 • 3 = 9 4 • 4 = 16 5 • 5 = 25 6 • 6 = 36 49, 64, 81, 100, 121, 144, ...
1. Simplify Find a perfect square that goes into 147.
2. Simplify Find a perfect square that goes into 605.
Simplify • . • . • . • .
How do you simplify variables in the radical? What is the answer to ? Look at these examples and try to find the pattern… As a general rule, divide the exponent by two. The remainder stays in the radical.
4. Simplify Find a perfect square that goes into 49. 5. Simplify
Simplify • 3x6 • 3x18 • 9x6 • 9x18
6. Simplify Multiply the radicals.
7. Simplify Multiply the coefficients and radicals.
Simplify • . • . • . • .
How do you know when a radical problem is done? • No radicals can be simplified.Example: • There are no fractions in the radical.Example: • There are no radicals in the denominator.Example:
8. Simplify. Uh oh… There is a radical in the denominator! Divide the radicals. Whew! It simplified!
9. Simplify Uh oh… Another radical in the denominator! Whew! It simplified again! I hope they all are like this!
10. Simplify Uh oh… There is a fraction in the radical! Since the fraction doesn’t reduce, split the radical up. How do I get rid of the radical in the denominator? Multiply by the “fancy one” to make the denominator a perfect square!
DELTAMATH DUE WEDNESDAY NIGHT 10/16 • Complete all sections in Deltamath before 10/16. This is 2 separate grades!