Question

How do you factor $ \left( {{m^3} + 64{n^3}} \right) $ ?

Answer 1

Hint: Given a polynomial is of degree $ 3 $ . Polynomials of degree $ 3 $ are known as Cubic polynomials. Cubic polynomials can be factored with the help of various techniques. One of the simplest and easiest methods is to factor the cubic polynomial using algebraic identities. We have many algebraic identities. So, we will have to choose the correct identity and implement it in the right way.

Complete step by step solution:
Given question requires us to factorize the polynomial $ \left( {{m^3} + 64{n^3}} \right) $
So, let the polynomial $ \left( {{m^3} + 64{n^3}} \right) $ be $ p(m) $ .
Then, $ p(m) = \left( {{m^3} + 64{n^3}} \right) $
The given polynomial is a cubic polynomial as the degree of the variable in the polynomial is $ 3 $ . Hence, the given polynomial should be factored using algebraic identities.
The polynomial is basically the sum of two cube terms. Hence, we can use algebraic identity $ {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right) $ to factorise the given polynomial.
In order to use the algebraic identity, we should convert the given polynomial into a form that would resemble the algebraic identity. So, we have,
 $ p(m) = \left( {{m^3} + 64{n^3}} \right) $
 $ \Rightarrow p(m) = {\left( m \right)^3} + {\left( {4n} \right)^3} $
Now, using the algebraic identity, $ {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right) $ , we get, $ $
 $ \Rightarrow p(m) = {\left( m \right)^3} + {\left( {4n} \right)^3} $
 $ \Rightarrow p(m) = \left( {m + 4n} \right)\left( {{m^2} - 4mn + {n^2}} \right) $
So, the factored form of the given polynomial $ \left( {{m^3} + 64{n^3}} \right) $ is $ \left( {m + 4n} \right)\left( {{m^2} - 4mn + {n^2}} \right) $ .
So, the correct answer is “ $ \left( {m + 4n} \right)\left( {{m^2} - 4mn + {n^2}} \right) $ .”.

Note: Such cubic polynomials and functions can be factored with the help of the algebraic identities with ease. Similar to the identity $ {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right) $ that is used in solving the question, we can also use the identity $ {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) $ in solving such question related to factorisation of cubic polynomials.