Proving equations!

[Synergy]

K I have two questions this time actually,

They are very similar, but here is first one:

Prove that (a^3 - b^3) is divisible by (a - b) and find other factors.

second question is:

Prove that (a^3 - b^3) is divisible by (a + b) and find other factors.

They look similar and I believe that the factors are like going to be similar for both but not sure then again! Please if anyone knows how to find, even one then from there I can work on other one!

Also can I ask how you people got so good in Calculus. Not that I suck I just get stuck sometimes on some tricky or complex problems! Please what did you do to get better! (like keep doing questions over, or do extra work aside teachers, P.S. I am still in high school!)

[zaza]

Start off with what you know. Assume that (a^3 - b^3) is divisible by (a-b).
Then there has to be a factor for this:

(a-b)(x) = (a^3 - b^3)

You know that the final result has to end up with a^3 in it, so you can guess that x begins with a^2.

(a-b)(a^2) = (a^3) -ba^2

Now you need to get rid of that (-ba^2) so what can you add on, bearing in mind that it's going to be multiplied by a (and then by -b, but forget that for the minute...)

Let's try (ba)...

(a-b)(a^2 + ba) = (a^3 - ba^2 + ba^2) - ab^2.


Now we've got our a^3 back again, but we've tacked (-ab^2) on the end. How can we get rid of this, again using some constant multiplied by a?

Let's try (b^2)...


(a-b)(a^2 + ba + b^2) = (a^3 - ba^2 + ba^2 - ab^2 + ab^2 -b^3)

...=(a^3 - b^3) as we wanted.



So another factor is (a^2 + ba + b^2).




Try it for the other equation...



zaza

[Synergy]

Thank you so much ZazA, do not know how to thank you enough!

[Synergy]

Hey i have another question, I get what we need and I have some idea of how to do it but i am not getting the whole picture somehow!

This is the question:

Let f(x) be a polynomial! When f(x) is divided by (x + 1) it has a remainder of 1. When f(x) is divided by (x - 1) the remainder is 3. Find the remainder when f(x) is divided by (x^2 - 1).

Now I know that when you multiply (x + 1) and (x - 1) we get (x^2 - 1). I am thinking you would plug in the information you have and then solve, for ex.

for (x + 1) and remainder is 1: I know that f(x) = (x+1) + 1/Q(x)

and

for (x - 1) and remainder is 3: I know that f(x) = (x-1) + 3/Q(x)

so if I would equate:

(x + 1)+ 1/Q(x) = (x - 1) + 3/Q(x) we would get this:

= (x^2 - 1) + 4/Q(x)

so would the remainder be 4 or am I lost somewhere?

[Synergy]

Start off with what you know. Assume that (a^3 - b^3) is divisible by (a-b).
Then there has to be a factor for this:

(a-b)(x) = (a^3 - b^3)

You know that the final result has to end up with a^3 in it, so you can guess that x begins with a^2.

(a-b)(a^2) = (a^3) -ba^2

I do not understand why when you multiple the left side with a^2 you get -ba^2 on right side! What happened to the b^3? Does it just get factored out so we can prove one equation is true to the other side?

[Synergy]

Ok no need to explain the one above this post I got it! Anyways I tried to do the second one:

(a^3 - b^3) = (a + b)(x) and this is how far I got before getting mixed up:

(a + b)(a^2 - ba + b^2) = (a^3 - ba^2 + ba^2 -ab^2 + ab^2 +b^3)

This issue is that last variable/coefficient, it is (+b^3) and not (-b^3)! Why so and what did I do wrong?

[zaza]

Maybe there is a typo in your original equation. Or maybe you are supposed to find out that it doesn't work.


Try filling in some numbers...after all if it works for a and b then it doesn't matter what the values are. You can easily show for a=5 and b=2 that a^3 - b^3 = 125 - 8 = 117. This is not divisible by 5+2 = 7.


zaza

[Synergy]

Hey zaza, I asked my teacher and seems that it was a typo! Meaning my answer was correct! Thanks again for the help!