Question

How do you expand ${\left( {x - y} \right)^3}$?

Answer 1

Hint: We have given an algebraic expansion and we have to expand it. Since it is a cubic binomial. So it can be written as ${a^3}$ equal to ${a^3} \times a$. Then the square term on the right-hand side can be written as ${a^2}$ is equal to $a \times a$. So the cubic binomial can be written as ${a^3}$ is equal to $a \times a \times a$. Now the first two terms are evaluated by distributive property. Then we add the like term. Then we evaluate the two terms again by distributive law. It is given the requested expansion form.

Complete Step by step Solution:

The given cubic binomial is ${\left( {x - y} \right)^3}$. We have to expand it.

We know that cubic binomial can be written as ${a^3} = {a^2} \times a$

So ${\left( {x - y} \right)^3} = {\left( {x - y} \right)^2} \times \left( {x - y} \right)$

Also the square binomial can be written as

${a^2} = a \times a$

So ${\left( {x - y} \right)^3} = {\left( {x - y} \right)^2} \times \left( {x - y} \right)$

Now we solve the first two terms by distributive law. By distributive law we have $\left( {a - b} \right) \times \left( {c - d} \right) = a \times \left( {c - d} \right) - b(c - d)$

So\[\left( {x - y} \right) \times \left( {x - y} \right) = x\left( {x - y} \right) - y(x - y)\]

\[ = {x^2} - xy - xy + {y^2}\]

Adding the like terms we get

\[\left( {x - y} \right) \times \left( {x - y} \right) = {x^2} - 2xy + {y^2}\]

Now the cubic binomial become

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

Again applying distributive law we get

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

$ = {x^3} - 2{x^2}y + x{y^2} - {x^2}y + 2x{y^2} - {y^3}$

Adding the like terms we get

\[\left( {{x^2} - 2xy + {y^2}} \right) \times (x - y) = {x^3} - 3{x^2}y = 3x{y^2} - {y^3}\]

This is the expanded form of ${\left( {x - y} \right)^3}$.

Note: A binomial is an algebraic expression in which there are two terms, for example $x + 3$ is a binomial algebraic expression. A cubic expression is the expression whose power is three. The sum of the cubic binomial is equal to the cube of the first terms, plus three times the first term by the square of second plus three terms the second term by the square of first term plus cube of the second term.