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.
 $$(a-b)\times (a^2+b^2-2ab)=a^3+a{b}^2-2a^{2}b-a^{2}b-b^3+2a{b}^2$$
Now let us group the like terms.
 $$(a-b)\times (a^2+b^2-2ab)=a^3+(a{b}^2+2a{b}^2)-(2a^{2}b+a^{2}b)-b^3$$
On simplifying this we get
 $$(a-b)\times (a^2+b^2-2ab)=a^3+(3a{b}^2)-(3a^{2}b)-b^3$$
Now let’s rearrange the above expression.
 $$(a-b)\times (a^2+b^2-2ab)=a^3-b^3+(3a{b}^2-3a^{2}b)$$
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.
 $$(a-b{)}^3=a^3-a^{2}b+a{b}^2-b^3-2a^{2}b+2a{b}^2$$
Let us group the terms like.
 $$(a-b{)}^3=a^3-(a^{2}b+2a^{2}b)+(a{b}^2+2a{b}^2)-b^3$$
On simplifying this we get
 $$(a-b{)}^3=a^3-(3a^{2}b)+(3a{b}^2)-b^3$$
Let us re-arrange the above expression.
 $$(a-b{)}^3=a^3-b^3-3a^{2}b+3a{b}^2$$
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 $$a^3-b^3-3ab(a-b)$$ .