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

Theory: