Solving Complex Math Problems: (-192x^2(4x-3)-4(27-64x^3))/(4x-3)^2

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Discussion Overview

The discussion revolves around simplifying a complex mathematical expression involving polynomial division and factoring. Participants explore various methods to simplify the expression and address challenges encountered during the process.

Discussion Character

  • Homework-related, Mathematical reasoning, Technical explanation

Main Points Raised

  • One participant expresses confusion about simplifying the expression and references a solution provided by Wolfram Alpha.
  • Another suggests expanding each bracket and simplifying the terms before factoring out (4x-3).
  • A participant shares their progress in simplifying the expression but struggles with factoring out (4x-3) from the resulting terms.
  • There is a suggestion to use synthetic or long division to proceed with the simplification.
  • One participant questions whether there is an easier method, noting that the problem is from a textbook and should not require long division.
  • Another participant points out that the term (27 - 64x^3) is a difference of cubes and suggests factoring it for potential cancellations.
  • A participant acknowledges the advice and reflects on the importance of experience in recognizing when to apply certain methods.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for simplification, with multiple approaches and some uncertainty about the necessity of long division versus factoring.

Contextual Notes

Some participants express uncertainty about the steps involved in simplification and the conditions under which different methods should be applied. There is also a lack of clarity on how to approach the problem effectively without prior experience.

Elpinetos
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I am currently stumped with a mental blackout, how do you simplify this again?

(-192x^2(4x-3)-4(27-64x^3))/(4x-3)^2

Wolfram Alpha spits out -32x-12, but how do you get there? :(
 
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Expand out each bracket first and then simplify the terms with x's in them.

2(x+1) = 2x+2

x^3 +x^2 +2x^3 + 2x^2 = 3x^3 + 3x^2

After doing that, try to factor out (4x-3)
 
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Now I have (-512x^3-576x^2-108)/(16x^2-24x+9), but how do I proceed?
I don't see how I can factor out (4x-3) out of this :/
 
Elpinetos said:
Now I have (-512x^3-576x^2-108)/(16x^2-24x+9), but how do I proceed?
I don't see how I can factor out (4x-3) out of this :/

Hi, try using synthetic or long division
 
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There must be an easier way, right? Oo

Since I got this from having to find the derivative of y=(27-64x^3)/(4x-3) and this is one of the basic problems in my textbook. I'm quite sure it doesn't involve long division oO
 
Well, you are the one that wanted to know how WA got its answer.

I think if you re-examine the numerator in you original expression, the term (27 - 64x^3) is the difference of two cubes. If you break this term into its factors, and then check for cancellations, it might be easier than expanding everything at first.
 
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Got it, thank you...
Fresh day, fresh mind

So simply make a polynomial devision BEFORE deriving... But how can I know that beforehand? :/
 
That's what the experience of doing a lot of math problems gives you.
 

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