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1) It's known that the product of two polinomials is a homogeneous polynomial. Prove that both factors are homogeneous polynomials too.

I proved that the product of homogeneous polynomials is homogeneous, but nothing works here :(

2) Prove that $x_1^3 + x_2^3 + x_3^3 $ is indecomposable.

Thanks in advance! Sorry for my English not being perfect, I'm non-native.

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We can prove (1) by contradiction. We will multiply together two polynomials, one of which is not homogeneous, and show that the product cannot be homogeneous Let $P_{1} = T_{1} + T_{2} + \cdots + T_{n}$ and $P_{2} = R_{1} + R_{2} + \cdots + R_{k}$, where $T_{i},R_{i}$ are nonzero terms. Then suppose that $P_{1}$ is not homogeneous, so that $\deg T_{1} \neq \deg T_{2}$. Then $$P_{1}P_{2} = R_{1}T_{1} + R_{1}T_{2} + \cdots$$

And since $R_{1},T_{1},T_{2}$ are nonzero, we know that $$\deg R_{1}T_{1} = \deg R_{1} + \deg T_{1} \neq \deg R_{1} + \deg T_{2} = \deg R_{1}T_{2}$$

So $P_{1}P_{2}$ was not homogeneous.

For part (2), it suffices to consider factoring your polynomial into homogeneous products. Consider all the possibilities and rule them out.

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1) Let $f,g$ be the two polynomials such that $h=fg$. Suppose $$f(x)=a_mx^m+\cdots+a_nx^n,$$ $$g(x)=a_px^p+\cdots+a_qx^q$$with $m>n,p>q$. Then $h(x)$ has the terms $a_ma_px^{m+p}$ and $a_na_qx^{n+q}$, and so it is not homogeneous.

2) By 1), if it is factorisable, then $x_1^3+x_2^3+x_3^3=(x_1+x_2+x_3)(x_1^2+x_2^2+x_3^2)$. But this is not possible as LHS$\neq$RHS.

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