Some deductions on cubic identities — lesson. Mathematics State Board, Class 8.

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1. Consider the standard identity I,

$(a+b)^3$

$=$

$a^3+3a^2b$

$+3ab^2+b^3$ .

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

Take the factor

$3ab$ from the last two terms of RHS.

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

Keep the required

$a^3+b^3$ in one side and the remaining in other side.

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

Taking the common factor

$(a+b)$ outside.

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

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

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

2. Consider the standard identity II,

$(a-b)^3$

$=$

$a^3-3a^2b$

$+3ab^2-b^3$ .

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

Take the factor

$3ab$ from the last two terms of RHS.

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

Keep the required

$a^3-b^3$ in one side and the remaining in other side.

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

Taking the common factor

$(a-b)$ outside.

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

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

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

Did you find an error?

4. Some deductions on cubic identities