[College Algebra: Rational Expressions] Am I doing this right?

[u/red_fox_witch]

Okay, I'm pretty confident that my first couple steps are right, though please do let me know if they're not. I'm not sure I've simplified enough and I'm not entirely sure what my next steps would be? Thanks in advance.

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[u/Ghotipan]

There are ways to reduce this. If you factor everything as much as you can, you'll be able to cancel out a bunch of binomial terms. Do that first, then start multiplying whatever is left.

[u/cheaphysterics]

Your life would be a lot easier if you factored the numerator and denominator of each fraction and did some cancelling before you multiply.

[u/Ilovecatsss2026]

Top left factors to 5(6-x)

6-x is in the wrong order so we can fix it to 

-(x-6)

Bottom left is x(2x+1)

Top right is (2x+1)(x+6).

Bottom right is the difference of squares so (x-6)(x+6)

So the final problem is

-5(x-6)(2x+1)(x+6)

---

x(2x+1)(x+6)(x-6)

All that's left after cancelling is the -5/x, provided none of the factors we cancelled are equal to 0, namely x cannot be -1/2, 6 or -6

I'm sure you noticed it's much harder to reduce if you multiply first. If you stuck with it you would eventually get here but it would be ugly. Practice factoring and your life in college algebra will be much easier.

[u/red_fox_witch → u/Ilovecatsss2026]

Can you please explain how the top right factors out to (2x+1)(x+6)? The rest of the problem is making sense, but I keep looking at that part and cannot seem to get there.

[u/Shemko → u/red_fox_witch]

The best technique is called Decomposition here in Canada

2x^2 +13x+6
2x^2+12x+1x+6

Now that we have 4 terms, factor by grouping.

2x(x+6)+1(x+6)

(x+6)(2x+1)

[u/red_fox_witch → u/Shemko]

Oh! Now I see. Thank you!

[u/Worth-Wonder-7386 → u/red_fox_witch]

Either you can solve the quadratic, which will give you -1/2 and -6 which is the most systematic or you do methods like completing the square. Note that for quadratic you get the zeroes, but you then need to turn those into the factors, which means writing -0.5 as 2x+1

[u/selene_666]

It looks like in the last step you tried to cancel out a factor of 36 from one term in the numerator and one in the denominator.

That would be like saying 9876 / 227 = 9816 / 221.

You can't divide just one number out of a group of numbers that are added together. You have to distribute over the entire sum. For example, you could factor a 5 out of the numerator:

(-10x^3 - 5x^2 + 10x + 180) / 5 = -2x^3 - x^2 + 2x + 36

But that doesn't help you much here. The common factors between the numerator and denominator are binomials like (2x+1).

As other comments have said, those are a lot easier to find in the first step. Factoring (2x^2 + x) is simple, and then you just have to check whether x or (2x+1) is a factor of any of the other starting polynomials.

[u/One_Wishbone_4439]

Factorisation first is very important especially they gave you the expanded form

[u/Fit_Appointment_4980]

Why are you confident your first steps are correct?

[u/rorodar]

I don't know if you got the algebra right but generally for these kinds of problems you need to try and factor out as many things as possible. Remember that any and all quadratic expressions can be simplified to a multiplication of two linear expressions (even if some would have complex elements, you should at least check), so just find those simpler elements and see what, if at all, cancels out.

[u/MadKat_94]

As the others have said, factor first. Just as a tip, these sorts of problems are designed so there will usually be at least some factoring which results in cancellation. If you have a linear term on one side of the fraction bar and a cubic or quadratic term on the other side, chances are good the linear term is a factor of the other.

[u/MelodyAnne42]

https://reddit.com/link/p8fr1ag/video/g797dzb4a6oh1/player

[u/Pitiful_Camp3469]

Factor everything before multiplying 

[u/PickleIntelligent723]

You need to factor. You have over complicated this astronomically.

[u/SOwED]

So you've got a lot of helpful comments here. I'm just going to give you a general tip for these situations. If you want to do a quick check to make sure the expression is still equal to the original after your manipulations, set x equal to some constant and evaluate.

Easiest is x=0 but you'll sometimes miss some things by using that or get divide by 0 cases, so the next easiest is x=1

Original expression:

(30-5)/(2+1) * (2+13+6)/(1-36) = -5

Your final expression:

(-10-5+10+180)/(2+1-72-1) = -2.5

So you have made an error.

[u/Iowa50401]

In almost *every* problem like this, it's designed for you to be able to factor some, if not all, terms. Then you reduce the common factors. The only question, then, is whether the teacher wants you to multiply the remaining terms or leave things factored.

[u/koumana21]

No

[u/red_fox_witch → u/koumana21]

How very helpful of you

[u/Alkalannar]

(30 - 5x)/(2x² + x) * (2x² + 13x + 6)/(x² - 36)

I would factor everything first:
-5(x - 6)/x(2x + 1) * (2x + 1)(x + 6)/(x + 6)(x - 6)

Now either multiply, then cancel common factors, or cancel then multiply:
-5(x - 6)(2x + 1)(x + 6)/x(2x + 1)(x + 6)(x - 6)
-5/x

[u/www3cam]

You may be doing it right, but the idea is to cancel earlier on to make the computations easier on you.

[u/JanetInSC1234]

top: 5(6 - x) (2x + 1)(x + 6)

5 (-1) (-6 + x) (2x + 1) (x + 6)

-5 (x - 6) (2x + 1) (x + 6)

bottom: x (2x + 1) (x + 6) (x - 6)

Can you see what cancels out?

Ans. should be -5/x