Prove using factor theorem.

Question

Using factor theorem, show that $a+b$,$b+c$ and $c+a$ are factors of

$(a + b + c)^3$ - $(a^3 + b^3 + c^3)$

How do we go about solving this ?

Thanks in advance !

Answer 1

To show, for example that $a+b$ is a factor of $p(a,b,c) = (a+b+c)^3 -(a^3 + b^3 + c^3)$, we consider $p$ as a polynomial in $a$, with constants $b$ and $c$, let's write $$ p_{b,c}(a) = p(a,b,c)$$ Now $a - (-b)$ is a factor of $p_{b,c}$ iff $-b$ is a root of $p_{b,c}$. We have \begin{align*} p_{b,c}(-b) &= (-b + b + c)^3 -\bigl( (-b)^3 + b^3 + c^3 \bigr) \\ &= c^3 - c^3\\ &= 0 \end{align*} So $$ a+b \mid p_{b,c}(a) $$

Answer 2

Let,$f(a)$=$(a+b+c)^3-(a^3+b^3+c^3)$

If $(a+b)$ is a factor,$f(-b)$ must be $0$ by factor theorem.

Putting $f(-b)$,we get,

$(-b+b+c)^3-(-b^3+b^3+c^3)$

$=c^3-c^3=0$.

Similarly prove the others.

Answer 3

If a+b is a Factor then -b is a root and vice versa. so just set a=-b and you get $((-b)+b+c)^3 - ((-b)^3+b^3+c^3) =c^3-c^3 = 0$. Therefor a+b is a Factor. Same for a+c and b+c.....