i need help it is so hard it ia from my textbook.

[ripe harbor]

Factorize the expression

[edgy gyro]

2[(a - b)^2 + (b - c)^2 + (c - a)^2].

wait i think i messed up

-((a^2 - b^2)^2) - (c^2(a^2 + b^2 + c^2))

this should be right

but I would wait for someone else to confirm

[ripe harbor]

Can u Factorize

[ripe harbor]

I think too

[rose dirge]

Motivation: Look at Heron's formula
@ripe harbor

https://en.wikipedia.org/wiki/Heron's_formula

[ripe harbor]

WHAT IS HERON

[rose dirge]

you pretty much find the answer in there

[ripe harbor]

https://cdn.discordapp.com/emojis/1001145428152172614.webp?size=48&quality=lossless

[ripe harbor]

IN THIS TRIANGLE

[rose dirge]

surprisingly yes

it's one of those expressions after you see it you recognise where you see it before

here's some motivation: Suppose you plug in a=x, b=y, c=x+y

[ripe harbor]

CAN U SOLVE IT AND EXPLAIN TO ME PLZ

[rose dirge]

what does the expression become

[ripe harbor]

OK THEN

[rose dirge]

(also note a triangle with sides x, y, x+y has area 0)

[ripe harbor]

HMMM

[rose dirge]

hence you can motivate from here (c-a-b) is one of the factors

[ripe harbor]

HOW

[rose dirge]

something like factor theorem

so when 2 is a zero of a polynomial, then x-2 is a factor

[ripe harbor]

OH YEA YEA

[rose dirge]

but more generalised, so we see c-a-b=0 results in the polynomial being 0, so that motivates c-a-b to be a factor

now use the symmetry to help you find other factors

[ripe harbor]

OH I THINK I KAINDA GET IT

[umbral cobalt]

.solved