Expanding cubic brackets (1 Viewer)

[whitnall8]

Is there a simple way to expand (a + b + c + d)^3?

 

[ranganerd]

not that i know of, just crunch the numbers

 

[pman]

they won't ask you to expand any cubics that can't be done using pascals triangle ot binomial thereom

 

[Drongoski]

For this you may use multinomial (instead of just 2-term = binomial) expansion.

But for say (a+b+c+d+e)^2 the following is both useful and easy to remember.

 

[pman]

but you won't get a cubic multinomial

 

[Trebla]

Write it as (a + b + c + d)(a + b + c + d)(a + b + c + d)

If you multiply each of the same letter three times as per each factor you get the terms:
a³ + b³ + c³ + d³

If you multiply TWO of the same letter but one is different from each factor, notice that that there are always three (3!/2!) ways to do so depending on which letter from each factor you multiply in a certain order (e.g. the term a²b comes from a.a.b, a.b.a or b.a.a):
3a²b + 3a²c + 3a²d +
3b²a + 3b²c + 3b²d +
3c²a + 3c²b + 3c²d +
3d²a + 3d²b + 3d²c

If you multiply all letters to be different from each factor, also notice that that there are always six (3!) ways to do so depending on which letter from each factor you multiply in a certain order so you get the terms (e.g. the term abc comes from a.b.c, a.c.b, b.a.c, b.c.a, c.a.b and c.b.a):
6abc +
6abd +
6acd +
6bcd

Adding all the terms of up gives:
a³ + b³ + c³ + d³ + 3a²b + 3a²c + 3a²d + 3b²a + 3b²c + 3b²d + 3c²a + 3c²b + 3c²d + 3d²a + 3d²b + 3d²c + 6abc + 6abd + 6acd + 6bcd

 

[life92]

This is going to end up pretty long regardless and Trebla's method is pretty good.

Alternatively, you can rewrite (a+b+c+d)^3 as [ (a+b) + (c+d) ] ^3
and then treat (a+b) and (c+d) as single terms.

So you'll end up with
(a+b)^3 + 3 (a+b)^2 (c+d) + 3(a+b) (c+d)^2 + (c+d)^3
Which you can then expand out again.

But regardless, a question like this will turn out to be very long and its unlikely to come out in an exam because of this and doesn't really test anything.

 

[cutemouse]

Yeah I think my 3U teacher taught me that method in Year 11 or something about expansions. (Trebla's method)

I had forgotten about that. Thanks for reminding me

 

[heteroskedastic]

just memorise some basic forms

(a + b = c)^3 - remember what it is

especially if you do ext2