 UPSKILL MATH PLUS

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

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, ${\left(x+y\right)}^{3}={x}^{3}+{y}^{3}+3\mathit{xy}\left(x+y\right)$.
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, ${\left(x-y\right)}^{3}={x}^{3}-{y}^{3}+3\mathit{xy}\left(x-y\right)$

Let us summarize the identities...

• $$(x+y)^3$$$$=$$$$x^3+3x^2y$$$$+3xy^2+y^3$$ or ${\left(x+y\right)}^{3}={x}^{3}+{y}^{3}+3\mathit{xy}\left(x+y\right)$
• $$(x-y)^3$$ $$=$$ $$x^3-3x^2y+3xy^2-y^3$$ or ${\left(x-y\right)}^{3}={x}^{3}-{y}^{3}+3\mathit{xy}\left(x-y\right)$

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