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Factorisation

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Express a4(b - c) + b4 (c - a) + c4 (a - b) as the product of four factors.

(b-c),(c-a) and (a-b) are products of this expression

OldTimer  Oct 20, 2017
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Express a4(b - c) + b4 (c - a) + c4 (a - b) as the product of four factors.

(b-c),(c-a) and (a-b) are products of this expression

$$\begin{array}{|rcll|} \hline a^4(b-c)+b^4(c-a)+c^4(a-b) &=& (b-c)(c-a)(a-b)\cdot x \\\\ x &=& \dfrac{a^4(b-c)+b^4(c-a)+c^4(a-b)}{(b-c)(c-a)(a-b)} \\\\ &=& \dfrac{a^4b-a^4c+b^4c-b^4a+c^4a-c^4b}{-a^2b+a^2c-b^2c+b^2a-c^2a+c^2b} \\\\ &=& \dfrac{ a^4·b - a^4·c - a·b^4 + a·c^4 + b^4·c - b·c^4 }{-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2} \\\\ \hline \end{array}$$

Polynom division after variable c:

$$\small{ \begin{array}{|rcll|} \hline && \text {in divisor term with max power of c is: } -ac^2 \\ && \text {current residue: } a^4·b - a^4·c - a·b^4 + a·c^4 + b^4·c - b·c^4 \\ && \text {in current residue term with max power of c: } ac^4 \\ && \text {quotient } \frac{ac^4}{-ac^2} = -c^2 \\ && \text {product } -c^2·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^2·b·c^2 - a^2·c^3 - a·b^2·c^2 + a·c^4 + b^2·c^3 - b·c^4 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^4·b - a^4·c - a^2·b·c^2 + a^2·c^3 - a·b^4 + a·b^2·c^2 + b^4·c - b^2·c^3 \\ && \text {in current residue term with next lower power of c is: } a^2·c^3 \\ && \text {quotient } \frac{a^2·c^3}{-ac^2} = -a·c \\ && \text {product } -a·c·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^3·b·c - a^3·c^2 - a^2·b^2·c + a^2·c^3 + a·b^2·c^2 - a·b·c^3 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^4·b - a^4·c - a^3·b·c + a^3·c^2 + a^2·b^2·c - a^2·b·c^2 - a·b^4 + a·b·c^3 + b^4·c - b^2·c^3 \\ && \text {in current residue term with next lower power of c is: } a·b·c^3 \\ && \text {quotient } \frac{a·b·c^3}{-a·c^2} = -b·c \\ && \text {product } -b·c·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^2·b^2·c - a^2·b·c^2 - a·b^3·c + a·b·c^3 + b^3·c^2 - b^2·c^3 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^4·b - a^4·c - a^3·b·c + a^3·c^2 - a·b^4 + a·b^3·c + b^4·c - b^3·c^2 \\ && \text {in current residue term with next lower power of c is: } a^3·c^2 \\ && \text {quotient } \frac{a^3·c^2}{-a·c^2} = -a^2 \\ && \text {product } -a^2·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^4·b - a^4·c - a^3·b^2 + a^3·c^2 + a^2·b^2·c - a^2·b·c^2 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^3·b^2 - a^3·b·c - a^2·b^2·c + a^2·b·c^2 - a·b^4 + a·b^3·c + b^4·c - b^3·c^2 \\ && \text {in current residue term with next lower power of c is: } a^2·b·c^2 \\ && \text {quotient } \frac{a^2·b·c^2}{-a·c^2} = -a·b \\ && \text {product } -a·b·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^3·b^2 - a^3·b·c - a^2·b^3 + a^2·b·c^2 + a·b^3·c - a·b^2·c^2 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } a^2·b^3 - a^2·b^2·c - a·b^4 + a·b^2·c^2 + b^4·c - b^3·c^2 \\ && \text {in current residue term with next lower power of c is: } a·b^2·c^2 \\ && \text {quotient } \frac{a·b^2·c^2}{-a·c^2} = -b^2 \\ && \text {product } -b^2·(-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2) = a^2·b^3 - a^2·b^2·c - a·b^4 + a·b^2·c^2 + b^4·c - b^3·c^2 \\ && \text {subtract product form current residue: } \\ && \text {current residue: } 0 \\\\ x&=& \dfrac{a^4·b - a^4·c - a·b^4 + a·c^4 + b^4·c - b·c^4 } {-a^2·b + a^2·c + a·b^2 - a·c^2 - b^2·c + b·c^2 } \\\\ &=& -a^2 - a·b - a·c - b^2 - b·c - c^2 \\ \hline \end{array} }$$

$$a^4(b-c)+b^4(c-a)+c^4(a-b) = (b-c)(c-a)(a-b)(-a^2 - a·b - a·c - b^2 - b·c - c^2)$$

heureka  Oct 20, 2017