Monday, Jan 7th Multiplying and Dividing Rational Expressions
Post date: Dec 18, 2013 9:52:20 PM
Notes:
What is Rational Expression?
An expression that is the ratio of two polynomials:
It is just like a fraction, but with polynomials.
How do we multiplying Rational Expressions?
Remember how you multiply regular fractions, you multiply across the top and bottom, and then you simply. This process works for rational expressions too.
Simplify the following expression:
![[ (7x^2) / 3 ] × [ 9 / (14x) ]](https://lh6.googleusercontent.com/S_SA2e1P_tO6exaAc92oN7SB9ty9IVt4TPnO7az3t1D7ogFKTVHupwzfUdvjvlIy-2A1LU-JIx4kMMlXHvfXa1iOCSwvXi00otNElzIevWY=w1280)
Simplify by cancelling off duplicate factors:
![[ (7x^2) / 3 ] × [ 9 / (14x) ] = (3x) / 2](https://lh4.googleusercontent.com/pHHoo0FtEjARUxlOnvdL1nbbt_1YFNrJNaQMAgxdU07CZonP5tke4N7FAMNUhRIEFRTIHB42-2R7qu1IdScsR_zigENg5F8pWLPrv4sF6P8=w1280)
Let's look at a harder one
Multiply and simplify the following expression:
![[(x^2 + 4x + 3)/(2x^2 - x - 10)] [(2x^2 + 4x^3)/(x^2 + 3x)] [x/(x^2 + 3x + 2)]](https://lh5.googleusercontent.com/W4CrrmxAeS5Yy_l5tGbi9dY73z-CVPE1SKwlKRuw-6GNXtWdDErS-6KqR--myVGoI_A1TFgJjjeEjG-s08mRUPKV4wGlvHaoj95C6eu6BzI=w1280)
Some students, when faced with this problem, will do something like this:
Can they really "cancel" like this? (Think "bleeding"...) Is this even vaguely legitimate? (Oozing...) Has this student done anything at all correctly? (Flopping, whimpering...) No, no, and no! You can't cancel terms, you can only cancel factors.
![(2x^2 + 4x^3)/[(2x - 5)(x + 2)^2]](https://lh4.googleusercontent.com/pnlcDv08SbHZ0UY84YQ9Av7Cls-in8NX2q5vyDXvoXGbTvlTFTvNmS1aXeitPkv_l1Is-Vu2eKHK3KNS2l3aNEYiAm0MPwudHXjxl4jmBao=w1280)
my answer, taking note of the trouble-spots (the division-by-zero problems) that I removed when I cancelled the common factors, is:
![[2x^2(1 + 2x) / (2x - 5)(x + 2)^2], x not equal to -1 or -3](https://lh6.googleusercontent.com/V9fHdNSspzx4rG_RBXwk3Etnh4dYnHuZ36YdTt0uX92dsotRN2i65aYZTb8gEWZXXZpaoIsZlRIGGW2Q6Hp1a8r8-UhRlXxoJavRILZIofA=w1280)
Now, let's look at dividing rational expressions
Division works the same way with rational expressions.
Perform the indicated operation:
![[ (x^2 + 2x - 15) / (x^2 - 4x - 45) ] ÷ [ (x^2 + x - 12) / (x^2 - 5x - 36) ]](https://lh4.googleusercontent.com/yWFBCw-tKD-iWplJkpFfUyQxP0YI6xD5H6Xw2OMxf1KaJi74BE1DDbL3XVlBlyBS1slZS5zr3HqiLZ6Lb8Ju5bZTgcm10psPsWAF5Z4xNM0=w1280)
To simplify this, first I'll flip-n-multiply. Then, to simplify the multiplication, I'll factor the numerators and denominators, and then cancel any duplicated factors. My work looks like this:
![[(x^2+2x-15)/(x^2-4x-45)] ÷ [(x^2+x-12)/(x^2-5x-36)] = [(x^2+2x-15)/(x^2-4x-45)] × [(x^2-5x-36)/(x^2+x-12)] = [(x+5)(x-3)(x-9)(x+4)]/[(x-9)(x+5)(x+4)(x-3)] = 1](https://lh3.googleusercontent.com/hqc5xd_KdbJX3uVnpIF4Am7SWUyP-4IBP5-S1THWzfQdaNCu2xfvZp7jISU9Gbg7IaFi288-3PkKsCCd7uaiEEvU0JriCm8V9H3BjPI9zQc=w1280)
Then the answer is: Copyright © Elizabeth Stapel 2003-2011 All Rights Reserved
Other Resources:
Rational Expressions : Multiplying and Dividing. Example 1
Rational Expressions: Multiplying and Dividing. Example 2
Rational Expressions: Multiplying and Dividing. Example 3
HW:
Worksheet 8.3 Multiplying and Dividing Rational Expressions pg 97-98
