Question

Expand each of the following using suitable identities ${\left( {p + 1} \right)^3}$.

Answer 1

Hint: We have to use the identity ${\left( {a + b} \right)^3}$ and expand the expression. ${\left( {a + b} \right)^3}$ denotes the cube of the sum of two numbers. The formula used to find the cube of a binomial is ${\left( {a + b} \right)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}$. The identity ${\left( {a + b} \right)^3}$ is obtained by multiplying $\left( {a + b} \right)$ three times. We can check that if we multiply \[\left( {a + b} \right)\] thrice then the product obtained is the same as ${\left( {a + b} \right)^3}$ i.e. ${a^3} + 3{a^2}b + 3a{b^2} + {b^3}$.

Complete step-by-step answer:
We have the expression ${\left( {p + 1} \right)^3}$. We need to expand the expression using a suitable identity.
${\left( {p + 1} \right)^3}$ denotes the cube of the sum of two numbers. Again, ${\left( {a + b} \right)^3}$ also denotes the cube of the sum of two numbers. So it is justified to use the identity ${\left( {a + b} \right)^3}$.
We use the identity ${\left( {a + b} \right)^3}$ to expand the given expression.
We replace $a = p$ and $b = 1$ in the identity ${\left( {a + b} \right)^3}$ to expand the expression ${\left( {p + 1} \right)^3}$.
Therefore, after replacing $a,b$ in the original expression we get:
${\left( {p + 1} \right)^3}$
$ = {p^3} + 3 \times {\left( p \right)^2} \times 1 + 3 \times p \times {\left( 1 \right)^2} + {\left( 1 \right)^3}$
$ = {p^3} + 3{p^2} + 3p + 1$
Therefore, we get the expression ${p^3} + 3{p^2} + 3p + 1$ after expanding the expression ${\left( {p + 1} \right)^3}$ using the identity ${\left( {a + b} \right)^3}$.
We can check if the result is correct or not by applying the general multiplication method for expanding the above expression.
Therefore we can multiply the expression $\left( {p + 1} \right)$ three times and then by performing the various arithmetic operations and by grouping the like terms we can find out the value of ${\left( {p + 1} \right)^3}$.

Note: Another formula for the identity ${\left( {a + b} \right)^3}$ is given by ${a^3} + {b^3} + 3ab\left( {a + b} \right)$. We can also use this identity to expand the required expression. Therefore, to expand the expression ${\left( {p + 1} \right)^3}$ we replace $a = p$ and $b = 1$ in the expression ${a^3} + {b^3} + 3ab\left( {a + b} \right)$ and we get:
${p^3} + {\left( 1 \right)^3} + 3 \times p \times 1 \times \left( {p + 1} \right)$
$ = {p^3} + 1 + 3p\left( {p + 1} \right)$
$ = {p^3} + 1 + 3{p^2} + 3p$
We get the same result as above. So using either of the formulas we can expand the given expression and both of them yield the same result.