Cubic identities — lesson. Mathematics State Board, Class 9.

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Let us derive the cubic identities with the help of known identitites.

Expansion of $(x+y)^3$:

Substitute $a = b = c = y$ in the identity $(x+a)(x+b)(x+c)$ $=$ $x^3$$+(a+b+c)x^2$$+(ab+bc+ca)x+abc$.

Consider the LHS, $(x+a)(x+b)(x+c)$.

$(x+a)(x+b)(x+c)$ $=$ $(x+y)(x+y)(x+y)$

$=$ $(x+y)^{3}$

Consider the RHS, $x^3$$+(a+b+c)x^2$$+(ab+bc+ca)x+abc$.

$x^3$$+(a+b+c)x^2$$+(ab+bc+ca)x+abc$ $=$ $x^3$$+(y+y+y)x^2$$+(yy+yy+yy)x+yyy$

$=$ $x^3$$+(3y)x^2$$+(y^2+y^2+y^2)x+y^3$

$=$ $x^3$$+3yx^2$$+3y^2x+y^3$

Thus, the identity is $(x+y)^3$ $=$ $x^3+3x^2y+3xy^2+y^3$.

The obtained cubic identity can also be rewritten as follows:

Consider the standard identity, $(x+y)^3$$=$$x^3+3x^2y$$+3xy^2+y^3$.

Take the factor $3xy$ from the middle two terms of RHS.

Thus,

{(x + y)}^{3} = x^{3} + y^{3} + 3 xy (x + y)

Expansion of $(x-y)^3$:

Replace $y$ by $-y$ in the cubic identity of $(x+y)^3$$=$$x^3+3x^2y$$+3xy^2+y^3$.

$(x+(-y))^3$ $=$ $x^3+3x^2(-y)+3x(-y)^2+(-y)^3$

$(x-y)^3$ $=$ $x^3-3x^2y+3xy^2-y^3$

Thus, the identity is $(x-y)^3$ $=$ $x^3-3x^2y+3xy^2-y^3$

The obtained cubic identity can also be rewritten as follows:

Consider the standard identity, $(x-y)^3$ $=$ $x^3-3x^2y+3xy^2-y^3$.

Take the factor $3xy$ from the middle two terms of RHS.

Thus,

{(x - y)}^{3} = x^{3} - y^{3} + 3 xy (x - y)

Let us summarize the identities...

$(x+y)^3$$=$$x^3+3x^2y$$+3xy^2+y^3$ or ${(x + y)}^{3} = x^{3} + y^{3} + 3 xy (x + y)$

$(x-y)^3$ $=$ $x^3-3x^2y+3xy^2-y^3$ or ${(x - y)}^{3} = x^{3} - y^{3} + 3 xy (x - y)$

Click here! to explore some examples on the expansion of cubic terms.

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3. Cubic identities