Question

Expand the following: $${\left( {3a + 2b} \right)^3}$$.
A.$$27{a^3} + 54{a^2}b + 6a{b^2} + 8{b^3}$$
B.$$27{a^3} + 54{a^2}b + 36a{b^2} + 8{b^3}$$
C.$$27{a^3} + 54{a^2}b + 36a{b^2} + 2{b^3}$$
D.$$9{a^3} + 54{a^2}b + 36a{b^2} + 8{b^3}$$

Answer 1

Hint: Here in this question, we need to expand the given algebraic expression by using a standard algebraic identity $${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$$ on comparing and substituting on further simplification by using a basic algebraic operations, we get the required solution.

Complete step-by-step answer:
The algebraic expression in mathematics is an expression which is made up of both variables and constants, along with algebraic operations like addition, subtraction, multiplication and division.
Algebraic Identities The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are used for the factorization of polynomials.
Consider the given algebraic expression:
$${\left( {3a + 2b} \right)^3}$$------(1)
Apply a standard algebraic identity $${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$$.
Where, $$a = 3a$$ and $$b = 2b$$.
On substituting in identity, then we have
  $$ \Rightarrow \,\,\,{\left( {3a + 2b} \right)^3} = {\left( {3a} \right)^3} + {\left( {2b} \right)^3} + 3\left( {3a} \right)\left( {2b} \right)\left( {3a + 2b} \right)$$
On multiplication, we have
$$ \Rightarrow \,\,\,{\left( {3a + 2b} \right)^3} = {\left( {3a} \right)^3} + {\left( {2b} \right)^3} + 18ab\left( {3a + 2b} \right)$$
Again, on multiplication, we have
$$ \Rightarrow \,\,\,{\left( {3a + 2b} \right)^3} = {\left( {3a} \right)^3} + {\left( {2b} \right)^3} + 54{a^2}b + 36a{b^2}$$
The cube number $${3^3} = 27$$ and $${2^3} = 8$$, then on simplification we get
$$\therefore \,\,\,{\left( {3a + 2b} \right)^3} = 27{a^3} + 8{b^3} + 54{a^2}b + 36a{b^2}$$
Hence, the expansion form of $${\left( {3a + 2b} \right)^3} = 27{a^3} + 8{b^3} + 54{a^2}b + 36a{b^2}$$.
Therefore, option (B) is the correct answer.
So, the correct answer is “Option B”.

Note: The algebraic identities play an important role in the mathematics curriculum and in mathematics in general. Student must knowing and recognising the algebraic expression with standard algebraic identities. It will also enable them to develop fluency when applying these procedures in algebraic manipulations and problem solving.