 So simplifying rational expressions is so important that we'll want to take a closer look at it And since we have to do it so often we want to see if there's a way that we can simplify that process So let's try to simplify this expression So remember always identify values that make the expression undefined We need to make sure the denominator isn't zero So we solve denominator equal to zero use your favorite method to find the solution And we get x equals four x equals one and We might observe x squared minus five x plus four is zero when x equals four or x equals one So we require x not equal to four and x not equal to one So now we can try to simplify But remember we can only cancel if both numerator and denominator are products and so this means we need to factor both numerator and denominator But factoring is hard Let's be clever about it We've already determined that x equals four and x equals one our solutions to x squared minus five x plus four equals zero And so the factor theorem tells us that the denominator must factor as x minus one root times x minus the other root Now remember the goal is to find a common factor in the numerator and the denominator And what this means is that if the numerator doesn't have a factor of x minus four or X minus one that we won't get any cancellation So in some sense, it's not worth factoring unless we know it has a factor of x minus four or x minus one So the real question isn't can we factor the numerator? the real question is does the numerator have a factor of x minus one or Does it have a factor of x minus four? So how can we tell if the numerator has a factor of x minus one or x minus four? This is where the rational root theorem comes in handy the rational root theorem guarantees that any Rational factors of four x squared minus 13 x plus nine must be of the form x minus a Where a is a divisor of nine over a divisor of four But four can't be written this way. So we know that x minus four isn't a factor Since one could be written this way x minus one might be a factor But we have to check and so we can use the remainder theorem and synthetic division to determine whether x minus one is a factor so we'll do that division and Since the remainder is zero then x minus one is a factor and our numerator is x minus one times four x minus nine and Now we have a common factor in numerator and denominator. We can cancel it out and get our simplified form There's one last important step simplification shouldn't change anything other than the appearance of our expression and so in our original expression we required x not be four and x not be one and so in our simplified expression We still require x not be four and x not be one and It's vitally important to keep this restriction. So we'll write that down as the last line of our simplification Nothing really changes if our expressions grow more complicated Bicking our favorite method of solving equations. We find that x squared minus x minus six equals zero when X equals three or negative two So we require x not be equal to three and x not be equal to two So let's factor since we know the roots of the denominator. We can factor it immediately Now we need to factor the numerator But remember we only care if x plus two or x minus three is a factor Any other factorization is unimportant. So the rational factors of x cubed minus eight x squared plus eleven x plus twelve Must be of the form x minus a where a is a divisor of twelve over a divisor of one Unfortunately, this means that x plus two and x minus three could be factors. We have to check them out So we'll check out x plus two using synthetic division we find and Since our remainder is fifty x minus negative two x plus two is not a factor We'll check x minus three. So again using synthetic division Since our remainder is zero We know that x minus three is a factor and so x cubed minus eight x squared plus eleven x plus twelve is x minus three times x squared minus five x minus four and So now we've written the numerator and denominator as products and we can drop out the common factors Can we go further? Well, we already know that x plus two is not a factor of the numerator So no further factorization is necessary What that means is if you're bored. It's a long weekend. You've binge watched everything you could possibly watch on Netflix Yeah, you could go ahead and try to factor x squared minus five x minus four But when you do you won't be able to obtain any further cancellations and so as far as the simplification of the problem is concerned we are done at this point when we write down the Original restrictions x cannot be equal to three x cannot be equal to negative two