Why is my teacher taking off points for not simplifying correctly?

[GreenPrint]

Help me "simplify"

Homework Statement

Ok my teacher has robbed me of points, seriously I'm tired of teachers stealing points from me because when your asked to simplify an expression your asking someone to do something that can't clearly be defined... you'll never ever come up with a clean cut definition of what it means to "simplify", people in like the sixth grade are taught to simplify by just combining terms together like

x - 4x = 3 -x

fine in the sixth grade if people can't "simplify" this to
-2x = 3
and solve for x
x = -3/2

sure take points off their grade but holy cow... in calculus a teacher is really going to take points off for not "simplifying completely" holy cow... we can't even define this term...

well you so I had to do a summer assignment and I handed it in and she stole points from me... I don't understand why... I know that there's something such as a ireducable simplification form or whatever you want to call it were you can't simplify furher... but please help me understand why this teacher is stealing points from me for no real reasonable reason at all here's the problems were I think taking points off from my grade is unreasonable...

Example 1: Simplify ((x+1)^3(x-2)+3(x-1))/((x+1)^4)
and I put for my answer
(x-2)/(x+1) + 3/(x^3 + 3x^2 + 3x + 1)
and she put a big orange x over it... so my question to you is why? What would be a more correct answer?

Example 2: Simplify (x^(1/2) - x^(1/3))/x^(1/6)
and I put for my answer
x^(1/3) - x^(1/6)
another big orange x... I have no clue why this is wrong... can someone please explain this to me... how can I write this more simple?

Thanks

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

Ok my teacher has robbed me of points, seriously I'm tired of teachers stealing points from me because when your asked to simplify an expression your asking someone to do something that can't clearly be defined... you'll never ever come up with a clean cut definition of what it means to "simplify", people in like the sixth grade are taught to simplify by just combining terms together like

x - 4x = 3 -x

fine in the sixth grade if people can't "simplify" this to
-2x = 3
and solve for x
x = -3/2

sure take points off their grade but holy cow... in calculus a teacher is really going to take points off for not "simplifying completely" holy cow... we can't even define this term...

well you so I had to do a summer assignment and I handed it in and she stole points from me... I don't understand why... I know that there's something such as a ireducable simplification form or whatever you want to call it were you can't simplify furher... but please help me understand why this teacher is stealing points from me for no real reasonable reason at all here's the problems were I think taking points off from my grade is unreasonable...

Example 1: Simplify ((x+1)^3(x-2)+3(x-1))/((x+1)^4)
and I put for my answer
(x-2)/(x+1) + 3/(x^3 + 3x^2 + 3x + 1)

The first fraction is correct, but the second one isn't. It looks like you canceled (x - 1) in the numerator with one of the factors of (x + 1)⁴ in the denominator. You can't do this.

And why did you choose to expand (x + 1)³ in the denominator?

and she put a big orange x over it... so my question to you is why? What would be a more correct answer?

Example 2: Simplify (x^(1/2) - x^(1/3))/x^(1/6)
and I put for my answer
x^(1/3) - x^(1/6)

This looks fine to me. You should ask her why she marked it wrong.

another big orange x... I have no clue why this is wrong... can someone please explain this to me... how can I write this more simple?

 

[eumyang]

Example 2: Simplify (x^(1/2) - x^(1/3))/x^(1/6)
and I put for my answer
x^(1/3) - x^(1/6)
another big orange x... I have no clue why this is wrong... can someone please explain this to me... how can I write this more simple?

This looks fine to me. You should ask her why she marked it wrong.

Hmm, would factoring out a $$x^{1/6}$$ be considered more simplified?
$$x^{1/3} - x^{1/6} = x^{1/6}(x^{1/6} - 1)$$
Maybe this is what the teacher was looking for?

 

[GreenPrint]

hmmm well interesting enough

((x+1)^3(x-2)+3(x-1))/((x+1)^4) = (x-2)/(x+1) + 3/(x^3 + 3x^2 + 3x + 1)
don't believe me check it plug in some random number for x and try both equations they are indeed equal to each other... I don't remember how I got that answer but it is a valid answer and I couldn't "simplify" it further

and yes I don't understand the second one either if she's really going to tell me that my answer is wrong because I didn't factor this
x^(1/3) - x^(1/6)

I will say that there's no need to factor... "simplify" is different from 'factor completely", I was always told in like sixth grade that "simplifying" meant to "do all the operations you can until you can't do any more operations" that's crap but it stuck with me...

like in the sixth grade
x + 2x
you wouldn't leave it like this because it's not as "simple" because you do the operation of adding the like terms and getting 3x... hmm so then (x+1)^2 is not as simple but yes from a factoring standpoint it's correct, well not really, but from a "simplification standpoint" it's not because the operation of multiplying it by itself can be done and your suppose to do all the operations you can until you can do no more...

hence it's the reason why I expanded (x+1)^3 and left it factored

well I hope you understand why she marked this stuff wrong and you I'm going to ask her tomorrow I just wanted to make sure I didn't sound like a complete idiot so if you guys can tell me what's wrong with my answers that would be great...

I just can't get over the fact that this is a calculus class and she's taking points off for this stuff lol...

 

[GreenPrint]

hmmm two results according to wolfram
http://www.wolframalpha.com/input/?i=simplify+(x%2B1)^2

 

[phyzguy]

((x+1)^3(x-2)+3(x-1))/((x+1)^4) = (x-2)/(x+1) + 3/(x^3 + 3x^2 + 3x + 1)
don't believe me check it plug in some random number for x and try both equations they are indeed equal to each other...

Nonsense! They are certainly not equal. If I plug in x=3, the LHS=35/128, and the RHS=19/64. Your teacher was correct in marking this one wrong. The other one is debatable.

 

[GreenPrint]

If we plug in six...

((6+1)^3(6-2)+3(6+1))/(6+1)^4 = .5801749271

(6-2)/(6+1) + 3/(6^3 + 3(6)^2 + 3(6) + 1) = .5801749271

?

Can you please show me what I did wrong

 

[GreenPrint]

Beats me but when I checked my work before I handed it in I always plug in six for x just to make sure and when I did this I got the same answers... and there exact values are the same I just checked on wolfram there both 199/343...

 

[GreenPrint]

well apparently people are reading this and don't know the answer and guess what I don't know either why is it that when I inserted six i got that the equations were equal it beats me if you could let me know that would be great as this was the reason why I thought they were equal... they are very much equal to each other I even graphed both equations ;O

 

[Mark44]

((x+1)^3(x-2)+3(x-1))/((x+1)^4) = (x-2)/(x+1) + 3/(x^3 + 3x^2 + 3x + 1)
don't believe me check it plug in some random number for x and try both equations they are indeed equal to each other... I don't remember how I got that answer but it is a valid answer and I couldn't "simplify" it further

That's not at all how it works. When you simplify an expression, you are writing it as another expression that is identically equal to the first expression. The fact that the two expressions happen to be equal for some arbitrary number is not really significant; the two expressions have to be equal for all numbers in the domain of both expressions.

As I said before, what it looks like you did was cancel an (x - 1) in the numerator of the second fraction with one of the (x + 1) factors in the denominator. That is wrong. Period.

and yes I don't understand the second one either if she's really going to tell me that my answer is wrong because I didn't factor this
x^(1/3) - x^(1/6)

I agree. If the only instructions given were to simplify the expression, then your answer is probably as good as the "correct" answer.

I will say that there's no need to factor... "simplify" is different from 'factor completely", I was always told in like sixth grade that "simplifying" meant to "do all the operations you can until you can't do any more operations" that's crap but it stuck with me...

like in the sixth grade
x + 2x
you wouldn't leave it like this because it's not as "simple" because you do the operation of adding the like terms and getting 3x... hmm so then (x+1)^2 is not as simple but yes from a factoring standpoint it's correct, well not really, but from a "simplification standpoint" it's not because the operation of multiplying it by itself can be done and your suppose to do all the operations you can until you can do no more...

hence it's the reason why I expanded (x+1)^3 and left it factored

(x + 1)^3 is factored. By expanding it, you are losing the factors. I can't see any good reason to expand (x + 1)^3 in this problem.

well I hope you understand why she marked this stuff wrong and you I'm going to ask her tomorrow I just wanted to make sure I didn't sound like a complete idiot so if you guys can tell me what's wrong with my answers that would be great...

 

[GreenPrint]

ok look
http://www.wolframalpha.com/input/?i=graph+((x%2b1)^3(x-2)%2b3(x-1))/((x%2b1)^4)++and+(x-2)/(x%2b1)+%2b+3/(x^3+%2b+3x^2+%2b+3x+%2b+1)&incParTime=true
go there as you can see they appear not to be equal to each other but when I plug in values were they appear not to be equal to each other in that graph like -2 on my calculator I got the same answer for both equations

 

[GreenPrint]

It's strange how when I put both equations into my graphing calculator they line up ontop of each other perfectly... and even calculating values at like -2 they are equal

try it for yourself

 

[GreenPrint]

Try and find a value for which there not equal to each other the guy earlier who said three was wrong check it... I really wish I remember what I did as this is kind of getting interesting here =)

 

[Mark44]

Look, I don't need a graphing calculator or Wolfram Alpha to be able to tell that these two expressions are not identically equal:
$$\frac{(x + 1)^3(x - 2) + 3(x - 1)}{(x + 1)^4}$$
and
$$\frac{x - 2}{x + 1} + \frac{3}{(x + 1)^3}$$

If you evaluate the first at x = 3, you get 70/256 = 35/128. If you evaluate the second at x = 3, you get 19/64. These are the same values that phyzguy got. That should be enough to convince you that your work is incorrect.

You have been told by at least three people (your teacher, phyzguy, and me) that your work is incorrect, and have been given solid reasons by phyzguy and me. What else do we have to do to convince you that your work is incorrect?

 

[GreenPrint]

ya i guess your right thanks =)

 

[Mark44]

If we plug in six...

((6+1)^3(6-2)+3(6+1))/(6+1)^4 = .5801749271

This should be ((6+1)^3(6-2)+3(6 - 1))/(6+1)^4
Note the red minus sign.

(6-2)/(6+1) + 3/(6^3 + 3(6)^2 + 3(6) + 1) = .5801749271

?

Can you please show me what I did wrong