Is a + b a Factor of a²(b + c) + b²(c + a) + c²(a + b) + 2abc?

[MorallyObtuse]

I tried, doubt I'm even close to correct. Show me where I went wrong or just guide me with the problem please.

Homework Statement

1.) Prove that a + b is a factor of a²(b + c) + b²(c + a) + c²(a + b) + 2abc and write down the other two factors.

2. The attempt at a solution

$$a^2(b - a) + b^2(-a + a) + (-a)^2(a + b) + 2ab(-a) = 0$$
$$a^2b - a^3 - b^2a + b^2a + a^2 + ab - 2ab = 0$$
$$a^2b - a^3 + a^2 - ab = 0$$$$a^2(-c + c) + (-c)^2(c +a) + c^2(a - c) + 2a(-c) = 0$$
$$-a^2c + a^2c + c^2 + ac + c^2a - c^3 - 2ac = 0$$
$$c^2 + c^2a - c^3 - ac = 0$$

$$(-b)^2(b + c) + b^2(c - b) + c^2(-b + b) + 2(-b)bc = 0$$
$$b^2 + bc + b^2c - b^3 - c^2b + c^2b - 2bc = 0$$
$$b^2 + b^2c - b^3 - bc = 0$$

 

[Mark44]

You're not given that a^2(b + c) + b^2(c + a) + c^2(a + b) + 2abc is equal to zero. All you're supposed to be doing is to rearrange what you have to put it into a factored form.

Having said that, how does the following expression relate to the expression above?
$$a^2(b - a) + b^2(-a + a) + (-a)^2(a + b) + 2ab(-a)$$

Also, are you sure that you have typed the problem exactly as it was given to you? I was able to factor the given expression into (b + c) times another factor, but I haven't been able to write it yet as (a + b) times another factor.

 

[HallsofIvy]

No, a+ b is a factor. Think of this as a polynomial in a with b and c constants. a+ b= a-(-b) will be a factor if and only if setting a= -b makes the polynomial equal to 0. And, of course, you can find the other factor by dividing by a+b.

 

[1/2"]

Hi!
Solution to this is quite simple..
1st open the brackets
:. you have
a²b+ab ²+b²c+c²b+a²c+c²a+2abc
=ab(a+b) +b²c+a²c+c²b+c²a+2abc=ab(a+b)+c ²(a+b)+b²c+a²c+2abc
= (ab+c ²)(a+b)+c(b ²+a ²+2ab)
=(ab+c ²)(a+b)+c(a+b)²
=(a+b)(c²+ab+ac+bc)
That's what I think.
(there could be some mistakes while typing in powers because I kinda get confused while typing them)
I hope this helps!

 

[MorallyObtuse]

Yes, I typed the question exactly as it was given.
I'm not sure how to 'prove'. The teacher keeps giving these proofs and I get baffled by them.

 

[MorallyObtuse]

a+ b= a-(-b) will be a factor if and only if setting a= -b makes the polynomial equal to 0.

a + b = 0, and so
a = -b or b = -a

 

[1/2"]

Hey what's the big deal then .
You "PROVE" it by giving the definition of factor and as it suits this condition :. You can say it is a factor of it.
See if it helps(?)