Question

Simplify this rational expression:

Answer 1

\[\frac{ x ^{3}+1 }{ x ^{3}-x ^{2}+x}\times \frac{ 6x }{ -54x-54 }\]

Answer 2

@Hero

Answer 3

I usually prefer rational equations, but I guess we can do this one

Answer 4

Hint:
First thing to notice is that in the first fraction, the numerator is a sum of cubes and the denominator, x is common to each term.

Answer 5

The sum of cubes formula is \((a^3 + b^3) = (a+b)(a^2 - 2ab + b^2)\)

Answer 6

Also, just for clarity, the numerator of the first fraction can be re-written as \(x^3 + 1^3\) since \(1^3 = 1\)

Answer 7

I don't understand the whole cubes formula.

Answer 8

It's just a formula. Keep that in mind. You need to replace the left side of that formula with the right side in your fraction.

Answer 9

How?

Answer 10

It's not really that difficult

\((a^3 + b^3) = (a+b)(a^2 - 2ab + b^2)\)
\((x^3 + 1^3) = (x + 1)(x^2 - 2x + 1)\)

Keep in mind that:
\(2x \dot\ 1 = 2x\)
\(1^2 = 1\)

Answer 11

It's just an annoying formula. I'm sorry that you do not understand

Answer 12

But anyway, if you can hold off your confusion for a bit longer, you'll understand that the numerator and denominator would look like this afterwards:

\[\large \frac{ (x + 1)(x^2 - 2x + 1)}{ x(x^2 - x + 1)} \times\ \frac{ 6x }{ -54x-54 }\]

Answer 13

The sum of cubes formula is a factoring method.

Answer 14

Just a "not so obvious" one. You have to push the "I believe" button in that regard.

Answer 15

The sum of cubes formula is like the slope-intercept formula. It's not something you necessarily have to understand. It's just something you memorize and apply when you recognize a sum of cubes such as \(a^3 + b^3\)

Answer 16

Oh my god, guess what I just realized.

Answer 17

I am so sorry, and I would hate me if I were you. But, the instructions were actually Multiply as indicated...

Answer 18

I already know that. It is obvious that you need to multiply the fractions since there is a multiplication symbol between the fractions, but you assume that you can just multiply as simple as that. In this case, we need to simplify the left fraction BEFORE multiplying it with the right fraction.

Answer 19

Ohh, okay. Well, I know how to factor the right side, and whatnot, I wasn't sure how to factor the left

Answer 20

The left fraction still needs to be simplified BEFORE you multiply them together.

Answer 21

The left fraction is not fully simplified yet

Answer 22

Ohh, okay.

Answer 23

You should be able to recognize a quadratic expression when you see one and then figure out if it can be factored. You do see quadratic expressions in the numerator and denominator, right?

Answer 24

Yeah

Answer 25

Okay, well, let me know the factored forms of each one.

Answer 26

maybe..\[x ^{2}(x+1)\] for the numerator?

Answer 27

You're losing me here. You need to show all of your work.

Answer 28

I guess I'm confused as to how I can factor the left

Answer 29

I guess I assumed you already knew how to factor

Answer 30

Well, I do...but not necessarily for the left side..

Answer 31

@GenGen96

Answer 32

yes

Answer 33

I'm sorry, I honestly don't mean to rush, but there is only this question nd one more and then I have to head off to bed, it's 11:25PM here

Answer 34

Okay, you said you factored it already so what do you have so far?

Answer 35

I have: \[\frac{ (x+1)(x ^{2}-+1) }{ x(x ^{2}-x+1) }\times \frac{ 6x }{ -54(x+1) }\]

Answer 36

From my end, you appear to have forgotten the 2x in the numerator of the first fraction. Is this what you have:

\[\large \frac{ (x + 1)(x^2 - 2x + 1)}{ x(x^2 - x + 1)} \times\ \frac{ 6x }{ -54x-54 }\]

Answer 37

i forgot the x, but why is there a 2x?

Answer 38

Oh, I see. I wrote down the formula wrong.

Answer 39

I did forget the x, though, so yes.

Answer 40

Okay, well after making corrections, this is what you should have:

\[\large \frac{ (x + 1)(x^2 - x + 1)}{ x(x^2 - x + 1)} \times\ \frac{ 6x }{ -54(x+1) }\]

Answer 41

And actually, we can immediately cancel something, can't we?

Answer 42

Do you see a factor of 1 that we can cancel?

Answer 43

Ohh, okay.

Answer 44

Since you think you have it figured out, let me know what the final result is.

Answer 45

do I cancel the x^2-x+1 too?

Answer 46

Yup. There are plenty of things that cancel

Answer 47

In a situation like this, what you want to do is organize everything so that you can keep track of what you need to cancel.

Answer 48

\[\frac{ 1 }{ x }\times \frac{ 6x }{ -54 }\]?

Answer 49

\[\large \frac{ 6x(x + 1)(x^2 - x + 1)}{ -54x(x+1)(x^2 - x + 1)} \]

Answer 50

Now it is more obvious what needs to be canceled

Answer 51

WHat about that x that was on the outside when everything was factored?

Answer 52

I included that as well, trust me

Answer 53

It's just hard for you to see it maybe

Answer 54

ohhh i see it

Answer 55

Okay, i got the answer. Thank you! Okay, last question..

Answer 56

You don't have the answer until you let me know what it is. It shouldn't be very difficult to type at all.

Answer 57

-1x/9x

Answer 58

The x's don't cancel?

Answer 59

Ohhhhh! Okay

Answer 60

so -1/9!

Answer 61

lol

Answer 62

Great job :D

Answer 63

:D Thank you! I'll start the question in a new thing and then close it! One second!

Answer 64

Okay, I mentioned you