Verify : (i) x³+ y ³= (x + y) (x²– xy + y²) (ii) x ³ – y³ = (x – y) (x² + xy + y² ).

Question

Find the square root of $4a^2 + 9b^2 + c^2 + 12ab - 6bc - 4ac$.

Answer 1

Step 1: Expand the right-hand side (RHS):

$ (x + y)(x^2 - xy + y^2) = x(x^2 - xy + y^2) + y(x^2 - xy + y^2) $

Step 2: Multiply each term:

$ x \cdot x^2 - x \cdot xy + x \cdot y^2 + y \cdot x^2 - y \cdot xy + y \cdot y^2 $

Step 3: Simplify terms:

$ x^3 - x^2y + xy^2 + x^2y - xy^2 + y^3 $

Step 4: Combine like terms ($ -x^2y + x^2y = 0 $, $ xy^2 - xy^2 = 0 $):

$ x^3 + y^3 $

✅ LHS = RHS, hence verified.

Step 1: Expand the RHS:

$ (x - y)(x^2 + xy + y^2) = x(x^2 + xy + y^2) - y(x^2 + xy + y^2) $

Step 2: Multiply each term:

$ x \cdot x^2 + x \cdot xy + x \cdot y^2 - y \cdot x^2 - y \cdot xy - y \cdot y^2 $

Step 3: Simplify terms:

$ x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3 $

Step 4: Combine like terms ($ x^2y - x^2y = 0 $, $ xy^2 - xy^2 = 0 $):

$ x^3 - y^3 $

✅ LHS = RHS, hence verified.

Solution and Explanation