Toán Lớp 9: `A=(a^3)/((a-b)(a-c))+(b^3)/((b-a)(b-c))+(c^3)/((c-a)(c-b))`

Question

Toán Lớp 9:  A=(a^3)/((a-b)(a-c))+(b^3)/((b-a)(b-c))+(c^3)/((c-a)(c-b))

hiennhi98 · Answer

$ begin{array}{l}  dfrac{{{a^3}}}{{ left( {a - b}  right) left( {a - c}  right)}} +  dfrac{{{b^3}}}{{ left( {b - c}  right) left( {b - a}  right)}} +  dfrac{{{c^3}}}{{ left( {c - b}  right) left( {c - a}  right)}}    =  dfrac{{{a^3} left( {c - b}  right) + {b^3} left( {a - c}  right) + {c^3} left( {b - a}  right)}}{{ left( {a - b}  right) left( {b - c}  right) left( {c - a}  right)}}    =  dfrac{{{a^3}c - {c^3}a + {b^3}a - {a^3}b + {c^3}b - {b^3}c}}{{ left( {a - b}  right) left( {b - c}  right) left( {c - a}  right)}}   TS:   {a^3}c - {c^3}a + {b^3}a - {a^3}b + {c^3}b - {b^3}c    = ac left( {{a^2} - {c^2}}  right) + {b^3} left( {a - c}  right) - b left( {{a^3} - {c^3}}  right)    = ac left( {a - c}  right) left( {a + c}  right) + {b^3} left( {a - c}  right) - b left( {a - c}  right) left( {{a^2} + ac + {c^2}}  right)    =  left( {a - c}  right) left( {ac left( {a + c}  right) + {b^3} - b left( {{a^2} + ac + {c^2}}  right)}  right)    =  left( {a - c}  right) left( {{a^2}c + a{c^2} + {b^3} - {a^2}b - abc - b{c^2}}  right)    =  left( {a - c}  right) left[ {ac left( {a - b}  right) + {c^2} left( {a - b}  right) - b left( {{a^2} - {b^2}}  right)}  right]    =  left( {a - c}  right) left( {a - b}  right) left( {ac + {c^2} - b left( {a + b}  right)}  right)    =  left( {a - c}  right) left( {a - b}  right) left( {ac + {c^2} - ab - {b^2}}  right)    =  left( {a - c}  right) left( {a - b}  right) left[ {a left( {c - b}  right) +  left( {c - b}  right) left( {c + b}  right)}  right]    =  left( {a - c}  right) left( {a - b}  right) left( {c - b}  right) left( {a + b + c}  right)    =  left( {c - a}  right) left( {b - c}  right) left( {a - b}  right) left( {a + b + c}  right)     Rightarrow A = a + b + c  end{array}$

cobelolen · Answer

Giải đáp:       Lời giải và giải thích chi tiết:   A=(a^3)/((a-b)(a-c))+(b^3)/((b-a)(b-c))+(c^3)/((c-a)(c-b))   ĐK:a  ne b  ne c   A=(a^3)/((a-b)(a-c))-(b^3)/((a-b)(b-c))+(c^3)/((a-c)(b-c))   A=(a^3(b-c))/((a-b)(a-c)(b-c))-(b^3(a-c))/((a-b)(a-c)(b-c))+(c^3(a-b))/((a-b)(a-c)(b-c))   A=(a^3(b-c)-b^3(a-c)+c^3(a-b))/((a-b)(a-c)(b-c))   A=((a-b)(a-c)(b-c)(a+b+c))/((a-b)(a-c)(b-c))   A=(a+b+c)