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.

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

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

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

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4. Some deductions on cubic identities

Theory: