Question

Докажите что если ab+bc+ac=0 то (a-b)(a-c)+(b-c)(b-a)+(c-a)(c-b)+a^2+b^2+c^2

Answer 1

-a4b+a4c+a3b2+2a3bc-2a3c2-2a2b3-a2b2c-a2bc2+a2c3+ab4+2ab3c-ab2c2+2abc3-ac4-b4c+b3c2-2b2c3+bc4 ————————————————————————————————————————————————————————————————————————————————————————————— (a-b)•(b-c)•(c-a)

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "c2"   was replaced by   "c^2".  2 more similar replacement(s).

Step by step solution :Skip Ad
Step  1  : c2 Simplify ————— c - a Equation at the end of step  1  : (a2) (b2) c2 ((—————•(a-c))+(—————•(b-a)))+(———•(c-b)) (a-b) (b-c) c-a Step  2  :Equation at the end of step  2  : (a2) (b2) c2•(c-b) ((—————•(a-c))+(—————•(b-a)))+———————— (a-b) (b-c) c-a Step  3  : b2 Simplify ————— b - c Equation at the end of step  3  : (a2) b2 c2•(c-b) ((—————•(a-c))+(———•(b-a)))+———————— (a-b) b-c c-a Step  4  :Equation at the end of step  4  : (a2) b2•(b-a) c2•(c-b) ((—————•(a-c))+————————)+———————— (a-b) b-c c-a Step  5  : a2 Simplify ————— a - b Equation at the end of step  5  : a2 b2•(b-a) c2•(c-b) ((———•(a-c))+————————)+———————— a-b b-c c-a Step  6  :Equation at the end of step  6  : a2•(a-c) b2•(b-a) c2•(c-b) (————————+————————)+———————— a-b b-c c-a Step  7  :Calculating the Least Common Multiple :

 7.1    Find the Least Common Multiple 

      The left denominator is :       a-b 

      The right denominator is :       b-c 

                  Number of times each Algebraic Factor
            appears in the factorization of:    Algebraic    
    Factor     Left 
 Denominator  Right 
 Denominator  L.C.M = Max 
 {Left,Right}  a-b 101 b-c 011

      Least Common Multiple: 
      (a-b) • (b-c) 

Calculating Multipliers :

 7.2    Calculate multipliers for the two fractions 


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = b-c

   Right_M = L.C.M / R_Deno = a-b

Making Equivalent Fractions :

 7.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)2   and  (y2+y)/(y+1)3  are equivalent as well. 

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

L. Mult. • L. Num. a2 • (a-c) • (b-c) —————————————————— = —————————————————— L.C.M (a-b) • (b-c) R. Mult. • R. Num. b2 • (b-a) • (a-b) —————————————————— = —————————————————— L.C.M (a-b) • (b-c) Adding fractions that have a common denominator :

 7.4       Adding up the two equivalent fractions 
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

a2 • (a-c) • (b-c) + b2 • (b-a) • (a-b) a3b - a3c - a2b2 - a2bc + a2c2 + 2ab3 - b4 ——————————————————————————————————————— = —————————————————————————————————————————— (a-b) • (b-c) (a - b) • (b - c) Equation at the end of step  7  : (a3b - a3c - a2b2 - a2bc + a2c2 + 2ab3 - b4) c2 • (c - b) ———————————————————————————————————————————— + ———————————— (a - b) • (b - c) c - a Step  8  :Calculating the Least Common Multiple :

 8.1    Find the Least Common Multiple 

      The left denominator is :       (a-b) • (b-c) 

      The right denominator is :       c-a 

                  Number of times each Algebraic Factor
            appears in the factorization of:    Algebraic    
    Factor     Left 
 Denominator  Right 
 Denominator  L.C.M = Max 
 {Left,Right}  a-b 101 b-c 101 c-a 011

      Least Common Multiple: 
      (a-b) • (b-c) • (c-a) 

Calculating Multipliers :

 8.2    Calculate multipliers for the two fractions 


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = c-a

   Right_M = L.C.M / R_Deno = (a-b)•(b-c)

Making Equivalent Fractions :

 8.3      Rewrite the two fractions into equivalent fractions

L. Mult. • L. Num. (a3b-a3c-a2b2-a2bc+a2c2+2ab3-b4) • (c-a) —————————————————— = ———————————————————————————————————————— L.C.M (a-b) • (b-c) • (c-a) R. Mult. • R. Num. c2 • (c-b) • (a-b) • (b-c) —————————————————— = —————————————————————————— L.C.M (a-b) • (b-c) • (c-a) Adding fractions that have a common denominator :

 8.4       Adding up the two equivalent fractions 

(a3b-a3c-a2b2-a2bc+a2c2+2ab3-b4) • (c-a) + c2 • (c-b) • (a-b) • (b-c) -a4b+a4c+a3b2+2a3bc-2a3c2-2a2b3-a2b2c-a2bc2+a2c3+ab4+2ab3c-ab2c2+2abc3-ac4-b4c+b3c2-2b2c3+bc4 ————————————————————————————————————————————————————————————————————— = ————————————————————————————————————————————————————————————————————————————————————————————— (a-b) • (b-c) • (c-a) (a-b) • (b-c) • (c-a) Final result : -a4b+a4c+a3b2+2a3bc-2a3c2-2a2b3-a2b2c-a2bc2+a2c3+ab4+2ab3c-ab2c2+2abc3-ac4-b4c+b3c2-2b2c3+bc4 ————————————————————————————————————————————————————————————————————————————————————————————— (a-b)•(b-c)•(c-a)

Processing ends successfully

Latest drills solved(-4,7)to(94,-55)(5)/(7)+(4)/(y)=38(x+8/9)-9a2/(a-b)(a-c)+b2/(b-c)(b-a)+c2/(c-a)(c-b) 

Answer 2

Для доказательства данного утверждения, раскроем скобки в выражении (a-b)(a-c)+(b-c)(b-a)+(c-a)(c-b). Получим:

(a-b)(a-c)+(b-c)(b-a)+(c-a)(c-b) = a^2 - ab - ac - ab + b^2 - bc + ac - bc + c^2 = a^2 + b^2 + c^2 - ab - ac - bc - ab - ac + ac - bc = a^2 + b^2 + c^2 - 2(ab + ac + bc)

Теперь подставим значение ab + bc + ac = 0 в полученное выражение:

a^2 + b^2 + c^2 - 2(ab + ac + bc) = a^2 + b^2 + c^2 - 2(0) = a^2 + b^2 + c^2

Таким образом, мы доказали, что если ab + bc + ac = 0, то (a-b)(a-c)+(b-c)(b-a)+(c-a)(c-b) равно a^2 + b^2 + c^2.