b^(2)c^(2)(b^(2)-c^(2))+c^(2)a^(2)(c^(2)-a^(2))+a^(2)b^(2)(a^(2)-b^(2))

asked by guest
on Nov 16, 2024 at 7:07 pm



You asked:

Evaluate the expression: b2c2(b2−c2)+c2a2(c2−a2)+a2b2(a2−b2){b}^{2} {c}^{2} \left({b}^{2} - {c}^{2}\right) + {c}^{2} {a}^{2} \left({c}^{2} - {a}^{2}\right) + {a}^{2} {b}^{2} \left({a}^{2} - {b}^{2}\right)

MathBot Answer:

Evaluated



b2c2(b2−c2)+c2a2(c2−a2)+a2b2(a2−b2)=a2b2(a2−b2)+a2c2(−a2+c2)+b2c2(b2−c2)\displaystyle {b}^{2} {c}^{2} \left({b}^{2} - {c}^{2}\right) + {c}^{2} {a}^{2} \left({c}^{2} - {a}^{2}\right) + {a}^{2} {b}^{2} \left({a}^{2} - {b}^{2}\right) = a^{2} b^{2} \left(a^{2} - b^{2}\right) + a^{2} c^{2} \left(- a^{2} + c^{2}\right) + b^{2} c^{2} \left(b^{2} - c^{2}\right)

Expanded

b2c2(b2−c2)+c2a2(c2−a2)+a2b2(a2−b2)=a4b2−a4c2−a2b4+a2c4+b4c2−b2c4{b}^{2} {c}^{2} \left({b}^{2} - {c}^{2}\right) + {c}^{2} {a}^{2} \left({c}^{2} - {a}^{2}\right) + {a}^{2} {b}^{2} \left({a}^{2} - {b}^{2}\right) = a^{4} b^{2} - a^{4} c^{2} - a^{2} b^{4} + a^{2} c^{4} + b^{4} c^{2} - b^{2} c^{4}

Factored

b2c2(b2−c2)+c2a2(c2−a2)+a2b2(a2−b2)=(a−b)(a−c)(a+b)(a+c)(b−c)(b+c){b}^{2} {c}^{2} \left({b}^{2} - {c}^{2}\right) + {c}^{2} {a}^{2} \left({c}^{2} - {a}^{2}\right) + {a}^{2} {b}^{2} \left({a}^{2} - {b}^{2}\right) = \left(a - b\right) \left(a - c\right) \left(a + b\right) \left(a + c\right) \left(b - c\right) \left(b + c\right)

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