Question

What is the formula of \[{(a - b)^3}\] ?

Answer 1

Hint: We can derive the formula with or without using a standard formula that is available. Even though we have a standard formula for \[{(a - b)^3}\] we can derive them by splitting them into its factors. Then multiplying those factors will give us the formula for \[{(a - b)^3}\] . We can also use any other standards formula to expand the factors.
Formula: The formula that we will be using for this is:
 \[{(a - b)^2} = {a^2} + {b^2} - 2ab\]
 \[{a^3} = a \times {a^2} = a \times a \times a\]

Complete step by step answer:
It is given that \[{(a - b)^3}\] we aim to find the formula for this term. First, we will split the term \[{(a - b)^3}\] into its factors.
Let us split \[{(a - b)^3}\] into its factors. Using the formula \[{a^3} = a \times {a^2} = a \times a \times a\] let’s split the given term by taking \[a\] as \[(a - b)\] .
 \[{(a - b)^3} = (a - b) \times {(a - b)^2}\]
Now we can use the formula \[{(a - b)^2} = {a^2} + {b^2} - 2ab\] to split the term \[{(a - b)^2}\] or we can just split that like \[{a^2} = a \times a\] by taking \[a\] as \[(a - b)\] .
Let us solve the problem in both ways.
First, let us use the formula \[{(a - b)^2} = {a^2} + {b^2} - 2ab\] to split the term \[{(a - b)^2}\] .
 \[{(a - b)^3} = (a - b) \times {(a - b)^2} = (a - b) \times ({a^2} + {b^2} - 2ab)\]
Now let us multiply the factors \[(a - b)\] & \[({a^2} + {b^2} - 2ab)\] term by term.
 \[(a - b) \times ({a^2} + {b^2} - 2ab) = {a^3} + a{b^2} - 2{a^2}b - {a^2}b - {b^3} + 2a{b^2}\]
Now let us group the like terms.
 \[(a - b) \times ({a^2} + {b^2} - 2ab) = {a^3} + (a{b^2} + 2a{b^2}) - (2{a^2}b + {a^2}b) - {b^3}\]
On simplifying this we get
 \[(a - b) \times ({a^2} + {b^2} - 2ab) = {a^3} + (3a{b^2}) - (3{a^2}b) - {b^3}\]
Now let’s rearrange the above expression.
 \[(a - b) \times ({a^2} + {b^2} - 2ab) = {a^3} - {b^3} + (3a{b^2} - 3{a^2}b)\]
Let’s take the term \[ - 3ab\] commonly out of the last two terms.
 \[(a - b) \times ({a^2} + {b^2} - 2ab) = {a^3} - {b^3} - 3ab(a - b)\]
Therefore, we get \[{(a - b)^3} = {a^3} - {b^3} - 3ab(a - b)\] .
Note: We can see that the formula can be derived by two methods: with standard formula or without standard formula. We will get the same answer for both methods.
Now let’s derive the formula without using the standard formula.
Consider the given term \[{(a - b)^3}\] .
Let’s split them into its factors using the formula \[{a^3} = a \times {a^2} = a \times a \times a\] .
 \[{(a - b)^3} = (a - b) \times (a - b) \times (a - b)\]
Now let’s multiply the first two factors.
 \[{(a - b)^3} = ({a^2} - ab - ab + {b^2}) \times (a - b)\]
On simplifying this we get
 \[{(a - b)^3} = ({a^2} + {b^2} - 2ab) \times (a - b)\]
Now let’s multiply the third term to the resultant.
 \[{(a - b)^3} = {a^3} - {a^2}b + a{b^2} - {b^3} - 2{a^2}b + 2a{b^2}\]
Let us group the terms like.
 \[{(a - b)^3} = {a^3} - ({a^2}b + 2{a^2}b) + (a{b^2} + 2a{b^2}) - {b^3}\]
On simplifying this we get
 \[{(a - b)^3} = {a^3} - (3{a^2}b) + (3a{b^2}) - {b^3}\]
Let us re-arrange the above expression.
 \[{(a - b)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}\]
Let’s take the term \[ - 3ab\] commonly out of the last two terms.
 \[{(a - b)^3} = {a^3} - {b^3} - 3ab(a - b)\]
Thus, we got the same answer for both methods. Therefore, the formula for the given term \[{(a - b)^3}\] is \[{a^3} - {b^3} - 3ab(a - b)\] .