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

UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

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