A Plus B Plus C Whole cube
Are you looking for A plus B plus C Whole cube? You can check the formulas of A plus B plus C Whole cube in three ways. We are going to share the (a+b+c)^3 algebra formulas for you as well as how to create (a+b+c)^3 and proof.
we can write: \((a+b+c)^3 = (a+b+c)(a+b+c)(a+b+c) \)
\(=>(a+b+c)^3 = (a+b+c)^2 (a+b+c) \) [we know that what is the formula of \( (a+b+c)^2 \)]
\(=>(a+b+c)^3 = (a^2+b^2+c^2 + 2ab +2bc +2ca) (a+b+c) \)
need too write in simple form of multiplication \(=>(a+b+c)^3 = a \times (a^2+b^2+c^2 + 2ab +2bc +2ca)\\ + b \times (a^2+b^2+c^2 + 2ab +2bc +2ca)\\ + c \times (a^2+b^2+c^2 + 2ab +2bc +2ca) \)
Simplify the all Multiplication one by one \((a+b+c)^3 = a \times (a^2+b^2+c^2 + 2ab +2bc +2ca)\\ + b \times (a^2+b^2+c^2 + 2ab +2bc +2ca)\\ + c \times (a^2+b^2+c^2 + 2ab +2bc +2ca) \)
\(=> (a+b+c)^3 = (a^3+ab^2+ac^2 + 2a^2b + 2abc + 2ca^2)\\ + b \times (a^2+b^2+c^2 + 2ab +2bc +2ca)\\ + c \times (a^2+b^2+c^2 + 2ab +2bc +2ca) \)
\(=> (a+b+c)^3 = (a^3+ab^2+ac^2 + 2a^2b + 2abc + 2ca^2)\\ + (a^2b+b^3+bc^2 + 2ab^2 + 2b^2c +2abc)\\ + c \times (a^2+b^2+c^2 + 2ab +2bc +2ca) \)
\(=> (a+b+c)^3 = (a^3+ab^2+ac^2 + 2a^2b + 2abc + 2ca^2)\\ + (a^2b+b^3+bc^2 + 2ab^2 + 2b^2c +2abc)\\ + (ca^2 + cb^2 + c^3 + 2abc + 2bc^2 + 2c^2a) \)
Arrage value according power and similear
\(=> (a+b+c)^3 = a^3 + b^3 +c^3 \\+ 6abc + 3a^2b+ 3ab^2 \\ + 3ac^2 + 3bc^2 +3b^2c + 3a^2c \)
\(=> (a+b+c)^3 = \\a^3 + b^3 +c^3 + 6abc+ 3ab (a+b) + 3ac (a+c) + 3bc (b+c) \)
(a+b+c)^3 Verifications
Need to verify \( (a+b+c)^3 \) formula is right or wrong. put the value of a = 1, b=2 and c=3
put the value of a and b in the LHS
\( (a+b+c)^3 = (1+2+3)^3 \)
\( 6^3 = 216 \)
put the value of a and b in the RHS
\(=> a^3 + b^3 +c^3 + 6abc+ 3ab (a+b) + 3ac (a+c) + 3bc (b+c) \)
\(=> 1^3 +2^3+3^3 +6 \times 1 \times 2 \times 3 + 3 \times 1 \times 2 (1+2) \\ + 3 \times 1 \times 3 (1+3) + 3 \times 2 \times 3 (2+3) \)
\(=> 1 +8+27 +36 + 6 (3) + 9 (4) + 18 (5) \)
\(=> 36 +36 + 18 +36 + 90 \)
\(=> 72 + 54 + 90 \)
\(=> 126 + 90 =216 \)
Therefore \( LHR = RHS \)
Proof Formula: \((a+b+c)^3 = \\a^3 + b^3 +c^3 + 6abc \\+ 3ab (a+b) + 3ac (a+c) + 3bc (b+c) \)
Summary (a+b+c)^3
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