Multiplying and dividing Rational functions with factoring first. There are to doing these problems: 1. Factoring And 2. Simplifying numerators with denominators This is an explosive lesson:

Sum and difference of cubes To factor the numerator and denominator you will use the factoring methods from the last lesson they include: Factor Common term Trinomial Grouping Difference of Squares, Sum and difference of cubes

Simplifying the numerator and denominator after factoring uses the following: 1.) if the numerator and denominator have something in common that is part of a monomial you can cancel out the part that is the same.Cancel things that are being multiplied. 2.) However to cancel out a polynomial from the numerator and denominator they must be the exact same. The 4 and 2 simplify separately the x3 and the x5 The binomial (x+2) in the numerator and denominator can cancel because the are the same however, the trinomials have a sign difference so they can not cancel at all!!

Lets start with simplifying a rational expression. Factor the numerator: Cancel out the common terms in the numerator and denominator. Original problem Factor the denominator:

The answer is…

Lets start with simplifying a rational expression. 3 Factor the numerator: Cancel out the common terms in the numerator and denominator. Original problem Factor the denominator:

Multiplying a rational expression. 2x(x-5) (x+3) (x+5)(x-5) 2x2 Factor all the numerators : Factor all the denominators:

The answer is… *** very important an x by itself can not cancel with an x that is being added or subtracted to a number

More Examples: Practice #15 4 𝑥 2 𝑥 2 −16 ∙ 𝑥 3 −64 2𝑥

Dividing Rational expressions You will flip over this one. Literally you flip the rational that comes AFTER the division sign then… The division becomes multiplication and you do the same thing as the last problems.

Change the division to multiplication and write the reciprocal of the fraction behind the operation. The x’s can cancel part of a monomial, the polynomials that are the same(x-4) and (x-2) can cancel as well

This same work also applies to complex fractions This same work also applies to complex fractions. Make them look the same.

Practice: #19) 4−𝑥 4+𝑥 4𝑥 𝑥 2 −16

Adding and Subtracting Rational Functions You still get to… … but only the denominator Factor Basically you find the Least common denominator (which will include factors) and multiply the numerators by the missing factors.

How do I get the LCD? First I find the smallest number that all the coefficients of any monomial will divide into evenly. Ex. 4 and 10 would be 20 because that is the smallest number that 4 and 10 divide into evenly. Next I take any base variable that is part of a monomial and in the denominator and use the one with the largest exponent Ex. x, x2 , and x3 the LCD is x3 Lastly any polynomial in the denominator in the denominator must be accounted for but not repeated unless it is repeated in the same fraction. Ex If these are the denominators of two fractions being added; (x + 2)(x + 2) and (x + 2)(x – 1). The LCD is (x + 2)2 (x – 1) because there are 2 (x + 2)’s in one denominator.

This rational needs to multiply by x. 1 x 3x2 3x(x-4) Multiply each numerator by what the denominator is missing. Factor the denominators. LCD: 3x2 (x-4) must contain all of the denominators but nothing extra

And the solution is… The denominator is the LCD and the numerator is all the numerators put together. If the numerator can factor do so and see if it will simplify with any factor in the denominator.

Practice #25