Understanding Rational Polynomials: Techniques and Simplification

Math A-100 IUSB Sec. 7.1 Detailed notes – Week 11 Rational Polynomials Rational polynomials (or as I call them, “Fractional polynomials”) are simply fractions with polynomials in the top and/or bottom. Many of the techniques used previously can be used here as well: 1) Evaluating for given value  plugging in a value for the Variable and simplifying Example 1: Simplify the rational polynomial undefined: 3x -8 for x = 6 x + 2 Plug in 6 for “x”: 3(6) - 8 then simplify: 18 - 8  10  5 (6) + 2 8 8 4 2) Exclusions: In some equations, there are values you cannot use for x: A fraction is undefined when the bottom = 0 , therefore we have to exclude any values that make the bottom of the fraction = 0 Example 1: Find the values for x that make this rational polynomial undefined: 3x -8 x + 2 The bottom of a fraction cannot = 0, so x + 2 cannot = 0  “x” cannot be -2 We can do the same with rational polynomials. Example 1: Evaluate 3x -8____ for x = 7 x 2 + 2x – 35 = 3(7) – 8_ __ = 21 – 8 = 13 (7) 2 + 2(7) – 35 49 +14 – 35 28 Example 2: Find the values for x that make this rational polynomial undefined 3x – 15___ x 2 + 2x – 35 Recall that “undefined” means the value make the bottom of the fraction = 0 So, factor and solve to find what values for x make x 2 + 2x – 35 = 0.

x 2 + 2x – 35 factors to (x + 7)(x – 5) giving you: (x + 7)(x – 5) = 0 Solving: x + 7 = 0 or x – 5 = 0 x = -7 or x = 5  excluded values for x = -7, 5  x = -7 or 5 make this rational polynomial undefined NOTE: Only the bottom of a fraction can make it undefined, so we can totally ignore the top of the fraction when looking for values that make a rational polynomial undefined. Simplifying rational polynomials Just like regular fractions, rational polynomials can be reduced/simplified. Recall when reducing fractions, you cancel out whatever goes into both the top number and the bottom . 24 reduces to 6 28 7 what you are actually doing is factoring both the top & bottom numbers, then cancelling whatever is the same in both. You can do the same with rational polynomials to simplify them. Example 1: Simplify 3x – 15___ x 2 + 2x – 35 factor both the top & bottom: 3x – 15___ becomes 3(x – 5)__ x 2 + 2x – 35 (x + 7)(x – 5) cancel out what is exactly the same in both top & bottom: becomes 3( x – 5)__ (x + 7) (x – 5) Leaving you with 3__ x + 7 NOTE: even though (x – 5) got cancelled out, it STILL has to be considered when looking for values to be excluded, because x = 5 will STILL make the original fraction undefined. Example 2: Simplify f(x) = (x – 4) 2 ___ x 2 - 12x + 32 factor both the top & bottom: f(x) = (x – 4) 2 ___ becomes (x – 4)(x – 4)__

x 2 - 12x + 32 (x - 8)(x - 4) cancel out what is exactly the same in both top & bottom: becomes (x – 4)( x – 4)__ (x - 8) (x – 4) Leaving you with x - 4__ x - 8 NOTE: you cannot reduce the 4 & 8, because each is part of a bigger polynomial: (x – 4) and (x – 8). You can only cancel parts that are exactly the same. TRICK: leave the ( ) in your answer  that may help you see if the parts are the same. Example 3: Simplify x 2 _- 36__ x 2 - 12x + 32 factor both the top & bottom: x 2 _- 36___ becomes (x – 6)(x + 6)__ x 2 - 12x + 32 (x - 8)(x - 4) There is nothing that is exactly the same in both top & bottom  does not simplify Leaving you with (x – 6)(x + 6)__ (x – 8)(x – 4) NOTE: when you leave it showing what the factoring is, we call this “factored form”. So, if your directions say to “Write your answer in factored form”, write it with all the pieces showing. Special cases We have some instances where it looks like a rational polynomial will not reduce, yet there are some “hidden factors” Example 1: Simplify x 2 + 3x - 28 16 - x 2 factor both the top & bottom: x 2 + 3x - 28___ becomes (x – 4)(x + 7)__ 16 - x 2 (4 - x)(4 - x) It LOOKS like there is nothing that is exactly the same in both top & bottom (x – 4) is NOT the same as (4 – x)  does not simplify Leaving you with (x – 4)(x + 7) (4 - x)(4 - x) BUT WAIT – there are some “hidden factors” here!! When you have only 2 terms, you can switch the order of terms by factoring out -1

16 - x 2 = -1( x 2 – 16 ) (try distributing the -1 and you get what you started) If we do this first, look how things change: f(x) = x 2 + 3x - 28___ becomes x 2 + 3x - 28 ___ becomes (x – 4)(x + 7)__ 16 - x 2 -1(x 2 - 16) -1(x + 4)(x – 4) Now there IS something to cancel: becomes ( x – 4)( x + 7)__ -1(x + 4)( x – 4) Leaving you with (x + 7) -1(x -4) NOTE: MML will NOT let you leave it like this: you need to get the -1 out of the bottom. So, multiply the top & bottom both by -1, just like you would to change a regular fraction, to cancel out the -1 in the bottom. (x + 7) becomes -1 (x + 7) becomes - 1 x - 7 -1(x -4) -1[ - 1(x -4)] x - 4 Multiplying and Dividing Rational polynomials Just like regular fractions, you can multiply and divide rational fractions. Multiplying Example 1: x 2 + 10x + 21_ * x 2 + 9x - 52 x 2 - 12x + 32 x 2 - 12x - 45 Unlike regular fractions, this can pretty nasty if you try to multiply the tops and then the bottoms, then reduce. x 2 + 10x + 21_ * x 2 + 9x - 52 = x 4 + 19x 3 – 121x 2 – 331x - 1092 x 2 - 12x + 32 x 2 - 12x – 45 x 4 – 24x 3 +131 x 2 + 416x - 1440 What on earth do you do this this?!?!?!?!?!? Recall that you can also cancel out parts of the fractions before you multiply. To do this, we need to factor all four parts to see what, if anything, we can cancel. x 2 + 10x + 21_ * x 2 + 9x - 52 x 2 - 12x + 32 x 2 - 12x - 45 becomes (x + 3)(x + 7)_ * (x – 4)(x + 13) (x – 8)(x – 4) (x – 15)(x + 3) NOW cancel ( x + 3 )(x + 7)_ * (x – 4)( x + 13) (x – 8 )(x – 4) (x – 15)( x + 3) Leaving you with (x + 7)_ * (x + 13) (x – 8) (x – 15)

We will write the answer in “factored form” (see above) to show that we cannot simply it further Answer: (x + 7)(x + 13) (x – 8)(x – 15) Example 2: x 2 – 36 _ * x 2 + 7x + 10 x 2 - 1x – 30 (x + 2) 2 becomes (x + 6)(x - 6)_ * (x + 2)(x + 5) (x – 6)(x + 5) (x + 2)(x + 2) cancelling (x + 6 )(x - 6)_ * (x + 2 ) (x + 5) (x – 6)( x + 5) (x + 2)( x + 2) Leaving you with (x + 6)_ or just x + 6 (x + 2) x + 2 NOTE: you can cancel both from one fraction to the other AND within the same fraction. You are simply reducing the fraction, like 6/8  3/4 Dividing We divide rational polynomials the same way we divide regular fraction: Flip the second fraction and multiply instead! Example 1: x 2 + 10x - 24_ divided by x 2 + 9x + 20 x 2 - 4x - 12 x 2 - 12x + 36 flip & multiply: x 2 + 10x - 24_ * x 2 – 12x + 36 x 2 - 4x - 12 x 2 + 9x + 20 becomes (x + 2)( x – 12) * (x - 6 )(x - 6) (x - 6 )( x + 2) (x + 4)(x + 5) Leaving you with (x -12)(x - 6) (x + 4)(x + 5)