Let $a,b$ and $c$ be real numbers.evaluate the following determinant: |$b^2c^2 ,bc, b+c;c^2a^2,ca,c+a;a^2b^2,ab,a+b$| Let $a,b$ and $c$ be real numbers. Evaluate the following determinant:
$$\begin{vmatrix}b^2c^2 &bc& b+c\cr c^2a^2&ca&c+a\cr a^2b^2&ab&a+b\cr\end{vmatrix}$$
after long calculation I get that the answer will be $0$. Is there any short processs? Please help someone thank you.

Imagine expanding along the first column. Note that the cofactor of $b^2c^2$ is $$(a+b)ac-(a+c)ab=a^2(c-b)$$ which is a multiple of $a^2$. The other two terms in the expansion along the first column are certainly multiples of $a^2$, so the determinant is a multiple of $a^2$. By symmetry, it's also a multiple of $b^2$ and of $c^2$. 
If $a=b$ then the first two rows are equal, so the determinant's zero, so the determinant is divisible by $a-b$. By symmetry, it's also divisible by $a-c$ and by $b-c$. 
So, the determinant is divisible by $a^2b^2c^2(a-b)(a-c)(b-c)$, a poynomial of degree $9$. But the detrminant is a polynomial of degree $7$, so it must be identically zero. 
