+0

Factorisation

+1
100
1
+25

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
Sort:

#1
+18827
+2

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

5 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details