Answer :
Answer:
Step-by-step explanation:
The algebraic identity a3 - b3 = (a - b)(a2 + ab + b2) is the two cubes' identity.
This can be proved by considering right hand side term.
(a - b)(a2 + ab + b2) = a(a2 + ab + b2) - b(a2 + ab + b2 )
=(a - b)(a2 + ab + b2) = a3 + a2b + ab2 - a2b - ab2 - b3 [by taking the common factor (a-b) out]
=a3 + a2b - a2b + ab2- ab2 - b3 [by bringing the like terms together and cancelling the possible terms like a2b and ab2 in the right hand side term]
= (a - b)(a2 + ab + b2) = a3 - b3 = L.H.S
Hence the equality is satisfied.
Answer:
Answer is given below in detail
Step-by-step explanation:
The equation \( a^3 + b^3 = (a+b)(a^2 + b^2 - ab) \) is a factorization identity known as the sum of cubes formula. Let's verify it step by step:
1. **Expand the Right-Hand Side (RHS):**
\[ (a+b)(a^2 + b^2 - ab) \]
Expand using distributive property:
\[ (a+b)(a^2 + b^2 - ab) = a(a^2 + b^2 - ab) + b(a^2 + b^2 - ab) \]
\[ = a^3 + ab^2 + a^2b + b^3 - ab^2 - a^2b - ab^2 + ab^2 \]
\[ = a^3 + b^3 \]
2. **Compare with Left-Hand Side (LHS):**
The left-hand side of the original equation is \( a^3 + b^3 \).
Since both the left-hand side (LHS) and the expanded right-hand side (RHS) are equal to \( a^3 + b^3 \), the identity \( a^3 + b^3 = (a+b)(a^2 + b^2 - ab) \) holds true.
This identity is useful in algebraic manipulations and factorizations involving cubes of sums and differences of variables.