Some deductions on cubic identities — lesson. Mathematics CBSE, Class 9.

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Let us derive some deductions on the learnt cubic identities.

Consider the identity \((a + b)^3\) \(=\) \(a^3\) \(+\) \(3a^2b\) \(+\) \(3ab^2\) \(+\) \(b^3\).

Factor out \(3ab\) from the last two terms of RHS.

{(a + b)}^{3} = a^{3} + b^{3} + 3 ab (a + b)

Retain \(a^3+b^3\) on one side and transform the remaining terms to the other side of the equation.

a^{3} + b^{3} = {(a + b)}^{3} - 3 ab (a + b)

Factor out \((a+b)\) from the RHS of the above equation and simplify.

a^{3} + b^{3} = (a + b) [{(a + b)}^{2} - 3 ab]

a^{3} + b^{3} = (a + b) [a^{2} + 2 ab + b^{2} - 3 ab]

a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})

Consider the identity \((a - b)^3\) \(=\) \(a^3\) \(-\) \(3a^2b\) \(+\) \(3ab^2\) \(-\) \(b^3\).

Factor out \(-3ab\) from the last two terms of RHS.

{(a - b)}^{3} = a^{3} - b^{3} - 3 ab (a - b)

Retain \(a^3-b^3\) on one side and transform the remaining terms to the other side of the equation.

a^{3} - b^{3} = {(a - b)}^{3} + 3 ab (a - b)

Factor out \((a-b)\) from the RHS of the above equation and simplify.

a^{3} - b^{3} = (a - b) [{(a - b)}^{2} + 3 ab]

a^{3} - b^{3} = (a - b) [a^{2} - 2 ab + b^{2} + 3 ab]

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

Did you find an error?

4. Some deductions on cubic identities