Description: MIT grad shows how to simplify a rational expression. To skip ahead: 1) for how to simplify an expression with DIFFERENCE of SQUARES factoring in it (x^2 49), skip to time 1:01. 2) for how to write the CONDITION or RESTRICTION, skip to time 3:52. 3) for how to simplify an expression with QUADRATIC(s) in it (like the x^2 3x + 2 trinomial), skip to time 6:12. 4) for an example with LARGER COEFFICIENTS and a NEGATIVE x^2 term (like 20x^2 + 14x + 12), skip to time 9:57. Nancy formerly of mathbff explains the steps:

For how to FACTOR QUADRATICS, jump to    • Factoring Quadratics... How? (NancyPi)  
For how to FIND the DOMAIN of a function, jump to    • How to Find the Domain of Any Functio...  
For more factoring examples in my video on how to solve quadratic equations by FACTORING, jump to:    • How to Solve Quadratic Equations by F...  

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There are three steps to simplify algebraic rational expressions:
1) Factor the numerator and denominator as much as possible.
2) Cancel factors that are the same on top and bottom.
3) Write a restriction ("condition") for any x values that used to make the denominator zero, before you cancelled factors. For instance: "x cannot equal 1".

1) FACTOR as much as you can: Factor the top and bottom as much as possible. In both the numerator and denominator, always check first if there's a number or a variable (x) that is in all the terms that you can pull out front. If you have something that looks like "x^2 minus a perfect square number", like x^2 9, you can factor using the difference of squares formula, into (x + 3)(x 3). If you have a difference of cubes (x^3 8) or sum of cubes (x^3 + 27), you can factor using the sum/difference of cubes formulas. If you have a quadratic expression on top or bottom that has three terms ("trinomial"), you can factor it using trial and error, or the "magic x" trick for factoring. To learn how to factor quadratics, check out my video on factoring.

2) CANCEL: Cross out any factors that are the same on top and bottom, and write what's left behind. When you've cancelled everything you can, you now have your simplified rational expression, or expression "reduced to lowest terms" or in "simplest form".

3) CONDITION: Is that all? To be technically correct, you should also write a note next to the simplified rational expression that states the restriction. You can check with your teacher to see if you’re required to write it for your class. But either way, the restriction is there, and the final answer isn’t really complete without this “condition”. All you need to do is: look at any factor(s) you cancelled in the denominator (ex: x7), see what x value used to make that factor equal to zero (7), and next to your answer write “x not equal to” that number (x not equal to 7).

We put the restriction, or condition, there at the end, because if we didn’t, the restricted x value would appear to be a valid x value for the expression, when it’s not. When we simplified the expression by cancelling the bottom factor, we got rid of the undefined, divisionbyzero problem for that xvalue, but in reality that xvalue is still not allowed. So to be totally correct, write the note that states this restriction next to your answer.

For how to find the HORIZONTAL ASYMPTOTE of a rational function, jump to    • How to Find the Horizontal Asymptote ...  

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Edited by Miriam Nielsen of zentouro:    / zentouro