Question

What is the $\left( {X - Y} \right)$ whole cube formula?

Answer 1

Hint: There can be only two ways to do this question. One way is that if we remember the formula which is very easy and the other way is the use of pascal's triangle. But it is recommended that we should learn all the formulas that we used in algebra.

Complete step by step solution:
The formula of $\left( {X - Y} \right)$ whole cube, i.e. ${\left( {X - Y} \right)^3}$ is given by:
$ \Rightarrow {\left( {X - Y} \right)^3} = {X^3} - {Y^3} - 3XY\left( {X - Y} \right)$
Or
$ \Rightarrow {\left( {X - Y} \right)^3} = {X^3} - {Y^3} - 3{X^2}Y + 3X{Y^2}$
Therefore, the required formula is ${\left( {X - Y} \right)^3} = {X^3} - {Y^3} - 3{X^2}Y + 3X{Y^2}$.
Additional information: As the power increases the expansion becomes lengthy and tedious to calculate. The Binomial Theorem is the method of expanding an expression which has been raised to any finite power. A binomial Theorem is a powerful tool of expansion, which has application in Algebra, probability, etc. Important points to remember about binomial expansion are-$\left( 1 \right)$The total number of terms in the expansion of \[{\left( {x + y} \right)^n}\;{\text{ }}are{\text{ }}\left( {n + 1} \right)\].$\left( 2 \right)$The sum of exponents of x and y is always n.$\left( 3 \right)$\[n{C_0},{{}^n}{C_1},{{}^n}{C_2},{\text{ }} \ldots {\text{ }}..,{{}^n}{C_n}\;\] are called binomial coefficients and also represented by \[{C_0},{\text{ }}{C_1},{\text{ }}{C_2},{\text{ }} \ldots ..,{\text{ }}{C_n}.\]$\left( 4 \right)$The binomial coefficients which are equidistant from the beginning and from the ending are equal, i.e. \[n{C_{0\;}} = {{}^n}{C_n},{{}^n}{C_{1\;}} = {{}^n}{C_{n - 1\;}},{{}^n}{C_{2\;}} = {{}^n}{C_{n - 2}}\;, \ldots ..{\text{ }}etc.\]

Note: The ${\left( {a - b} \right)^3}$ formula is used to find the cube of a binomial. This formula is also used to factorize some special types of trinomials. This formula is one of the algebraic identities. The ${\left( {a - b} \right)^3}$ formula is the formula for the cube of the difference of two terms. We can also do this by binomial expansion but this is a little lengthy approach. It’s general formula is \[{\left( {x{\text{ - }}y} \right)^n}\; = \sum\limits_{r=0}^n{{}^n}{C_r}\;{x^{n-r}}\cdot{\left({-y} \right)^r}\]. This formula is used to calculate the cube of the difference of two terms very easily and quickly without doing complicated calculations.