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The expression a(b - c)^3 + b(c - a)^3 + c(a - b)^3 can be factored into the form (a - b)(b - c)(c - a) p(a,b,c), for some polynomial p(a,b,c). Find p(a,b,c)

Aug 28, 2019

#1
+25267
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The expression $$a(b - c)^3 + b(c - a)^3 + c(a - b)^3$$ can be factored into the form

$$(a - b)(b - c)(c - a) p(a,b,c)$$, for some polynomial $$p(a,b,c)$$

Find p(a,b,c)

$$\begin{array}{|rcll|} \hline a(b - c)^3 + b(c - a)^3 + c(a - b)^3 &=& -a^3b + a^3 c + a b^3 - a c^3 - b^3 c + b c^3 \\ (a - b)(b - c)(c - a) &=& -a^2b + a^2 c + a b^2 - a c^2 - b^2 c + b c^2 \\ \hline -a^3b + a^3 c + a b^3 - a c^3 - b^3 c + b c^3 &=& (-a^2b + a^2 c + a b^2 - a c^2 - b^2 c + b c^2)(a+b+c) \\\\ &=& -a^3b + a^3 c + a b^3 - a c^3 - b^3 c + b c^3 \\ && +a^2b^2-a^2b^2 \\ && +a^2c^2-a^2c^2 \\ && +b^2c^2-b^2c^2 \\ && +a^2bc-a^2bc \\ && +b^2ac-b^2ac \\ &&+c^2ab-c^2ab \\\\ \mathbf{a(b - c)^3 + b(c - a)^3 + c(a - b)^3} &=&\mathbf{ (a - b)(b - c)(c - a)(a+b+c) } \\ \hline \end{array}$$

$$\mathbf{p(a,b,c) = a+b+c}$$

Aug 29, 2019
edited by heureka  Aug 29, 2019

#1
+25267
+3

The expression $$a(b - c)^3 + b(c - a)^3 + c(a - b)^3$$ can be factored into the form

$$(a - b)(b - c)(c - a) p(a,b,c)$$, for some polynomial $$p(a,b,c)$$

Find p(a,b,c)

$$\begin{array}{|rcll|} \hline a(b - c)^3 + b(c - a)^3 + c(a - b)^3 &=& -a^3b + a^3 c + a b^3 - a c^3 - b^3 c + b c^3 \\ (a - b)(b - c)(c - a) &=& -a^2b + a^2 c + a b^2 - a c^2 - b^2 c + b c^2 \\ \hline -a^3b + a^3 c + a b^3 - a c^3 - b^3 c + b c^3 &=& (-a^2b + a^2 c + a b^2 - a c^2 - b^2 c + b c^2)(a+b+c) \\\\ &=& -a^3b + a^3 c + a b^3 - a c^3 - b^3 c + b c^3 \\ && +a^2b^2-a^2b^2 \\ && +a^2c^2-a^2c^2 \\ && +b^2c^2-b^2c^2 \\ && +a^2bc-a^2bc \\ && +b^2ac-b^2ac \\ &&+c^2ab-c^2ab \\\\ \mathbf{a(b - c)^3 + b(c - a)^3 + c(a - b)^3} &=&\mathbf{ (a - b)(b - c)(c - a)(a+b+c) } \\ \hline \end{array}$$

$$\mathbf{p(a,b,c) = a+b+c}$$

heureka Aug 29, 2019
edited by heureka  Aug 29, 2019