Finding Sum/Diff. Formulas: Quadratic to Higher Exponents

[EProph]

For the difference of two squares you can factor using: $$a^2 - b^2 = (a + b)(a - b)$$.

And for the sum and difference of two cubes:
$$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$$
$$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$$

How are these formulas found...?
Can it be done with basic algebra (am I missing something really simple)?
For example, it is possible to take $$ax^2 + bx + c = 0$$ and solve for x to get the quadratic formula. But how do you rework $$a^2 - a^2$$, or $$a^3 \pm b^3$$ into their respective formulas? What about with higher exponents (i.e. $$a^4 \pm b^4$$ or or $$a^5 \pm b^5$$)?

I realize that the resulting factors must, when multiplied, have middle terms that will completely cancel each other out leaving only $$a^n \pm b^n$$, but how do you work backwards to find the middle terms of the factors?

-EP

 

[Oggy]

You go the other way, i.e.
$$a^{2}-b^{2}=a^{2}-ab+ab-b^{2}=a(a-b)+b(a-b)=(a-b)(a+b)$$
$$a^{3}-b^{3}=a^{3}+a^{2}b-a^{2}b+ab^{2}-ab^{2}-b^{3}=a^{2}(a-b)+ab(a-b)+b^{2}(a-b)=(a-b)(a^{2}+ab+b^{2})$$...

 

[EProph]

I understand that you can take the formula and work it backwards to find the middle terms. But what if you don't have the formula? Someone had to find the formula, so how did they find it?

Forgive me if I'm being obtuse, but what mathematical process is used to get from $$a^3 - b^3$$ to $$a^3-a^2b+ab^2+a^2b-ab^2+b^3$$

I'm working under the assumption that the end formula is not necessary in mathematically determining the middle terms... am I incorrect?
I ask because without the formula as a reference, it doesn't seem like it is possible to find the middle terms without a bit of trial and error. Is this the case? And what about cases where the binomial can't be factored? (i.e. $$a^2 + b^2$$).

-EP

 

[dextercioby]

Did u ever hear of polynomials' division...?The polynomial $$P(a)=a^{n}-b^{n}$$ has the root a=b...Agree.Then please divide the 2 polynomials...

As for his brother,$$P(a)=a^{n}+b^{n}$$,well,it can't be factored into the reals...Not really for doing integrals in the reals.

Daniel.

 

[EProph]

Ah, found what I was looking for!

http://mathworld.wolfram.com/BinomialNumber.html

Thanks,
-EP

 

[dextercioby]

Hold it,what have Binomial numbers (i hope they make up Pascal's triangle) with [itex]a^{n}-b^{n} [/tex]...?Binomial numbers come from $(a+b)^{n}$,which totally another animal...<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f609.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":wink:" title="Wink :wink:" data-smilie="2"data-shortname=":wink:" /><br /> <br /> Daniel.[/itex]

 

[hypermorphism]

I understand that you can take the formula and work it backwards to find the middle terms. But what if you don't have the formula? Someone had to find the formula, so how did they find it?

Forgive me if I'm being obtuse, but what mathematical process is used to get from $$a^3 - b^3$$ to $$a^3-a^2b+ab^2+a^2b-ab^2+b^3$$

Creativity aided by experience.

 

[HallsofIvy]

Forgive me if I'm being obtuse, but what mathematical process is used to get from $$a^3 - b^3$$ to $$a^3-a^2b+ab^2+a^2b-ab^2+b^3$$

-EP

There is NO process that will get you from $$a^3 - b^3$$ to $$a^3-a^2b+ab^2+a^2b-ab^2+b^3$$

Perhaps you are thinking of
$$a^3- b^3= (a- b)(a^2+ ab+ b^2)$$

The way that was found was exactly what you were told before: people noticed that multiplying (a-b)(a²+ ab+ b² gave you
a³- b³

It is true that aⁿ- bⁿ= (a-b)(a^n-1+ a^n-2b+ ... + ab^n-1+ bⁿ.

IF n IS ODD then aⁿ+ bⁿ= (a+b)(a^n-1- a^n-2b+ ... - ab^n-1+ bⁿ. But aⁿ+ bⁿ for n even, such as a²+ b² or a⁴+ b⁴ cannot be factored (in terms of real numbers).