Questions About Factoring Patterns

[doughishere]

So i was doing some binomial theorem work this morning...basically just looking at the various rules and then expanding them to like the simple powers....basically i went up to the power of 5 writing every step down.....anyways

Is there a catch all formula for \(\displaystyle a^n+b^n\) i see one for a \(\displaystyle a^n-b^n\) and one if n is odd but no even....whats the thought behind the even powers?...see attached image with my highlights...



So kinda just realized as im typing this out the even powers of n you need an imaginary number to factor it....is there a catch all rule for evens? like say i wanted to find \(\displaystyle a^6+b^6\)...what would it look like if i made


14. \(\displaystyle a^4+b^4 =\)
16. \(\displaystyle a^6+b^6 =\)

 

[doughishere]

I think i found it.....my next question is whats the sequence of the addition and subtraction symbols in the second factor be


For instance the first one would be - +...?...+- for n

the second would be -+...?...-+ for n

what would the sequence be if we went to n+1?

 

[mmm4444bot]

So i was doing some binomial theorem work this morning …

This statement could refer to any number of different things.

… basically just looking at the various rules and then expanding them …

You're expanding rules?

I'm not sure what "the various rules" are (that you're looking at), associated with the Binomial Theorem.

Please be more specific; try not to use unreferenced pronouns.

Is there a catch all formula for \(\displaystyle a^n + b^n\)

If you're talking about factoring that binomial expression, the answer is, "No". There are different formulas for different cases (see the posted link, below.)

Also, the Binomial Theorem is used for expanding powers of binomial expressions.

EG: (a - b)^4

The Binomial Theorem explains why the expanded form is:

a^4 - 4a^3 b + 6a^2 b^2 - 4a b^3 + b^4

Yet, your examples deal with factoring binomial expressions.

EG: a^4 - b^4 = (a + b)(a - b)(a^2 + b^2)

Are you looking at some on-line source showing how to use the Binomial Theorem to factor binomial expressions? I would like to ensure that you're not confusing two topics (expansion vs factoring OR "binomial expression of like powers" vs "powers of binomial expressions").

what would it look like if i made

14. \(\displaystyle a^4+b^4 =\)

16. \(\displaystyle a^6+b^6 =\)

a^4 + b^4 = (a^2 + √2 ab + b^2)(a^2 - √2 ab + b^2)

a^6 + b^6 = (a^2 + b^2)*(a^4 - a^2*b^2 + b^4)

You may find this page helpful. Scroll down to Section 2.3 (subtitled as "Recognizable patterns"). :cool: