Question

How do you factor completely \[4{x^3} - 36x\]?

Answer 1

Hint: Here, we are required to factorize the given polynomial. We will first factor out the common term from each term of the given expression. We will then get two factors as a product of a linear polynomial and a quadratic polynomial. We will then factorize the quadratic polynomial by using a suitable algebraic identity to get the required factors of the given polynomial.

Formula Used:
\[\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)\]

Complete step-by-step answer:
In order to write the given cubic equation \[4{x^3} - 36x\] in factored form, we will use the direct method.
In this method, we directly take out the greatest factor which is common from the polynomial and leave the rest of the terms inside the bracket. The terms inside the bracket become one factor and the common terms that are taken out to become the other factor of the given polynomial.
The given polynomial is: \[4{x^3} - 36x\]
The greatest common factor is \[4x\]
Hence, taking it out from the bracket, we get,
\[4{x^3} - 36x = 4x\left( {{x^2} - 9} \right)\]
Now, as we know that 9 is the square of 3
Thus, using the identity \[\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)\]in the bracket, we get,
\[4{x^3} - 36x = 4x\left( {{x^2} - {3^2}} \right) = 4x\left( {x - 3} \right)\left( {x + 3} \right)\]

Hence, clearly, the factors of the given polynomial are \[4x,\left( {x - 3} \right),\left( {x + 3} \right)\]
Therefore, we have factored completely the given cubic equation.

Note:
An equation is called a cubic equation if it can be written in the form of \[a{x^3} + b{x^2} + cx + d = 0\] where \[a,b,c,d\] are real numbers and \[a \ne 0\] , as it is the coefficient of \[{x^3}\] and it determines that this is a cubic equation. Also, the power of a cubic equation will be 3 as it is a ‘cubic equation’. Also, a cubic equation will give us three roots whereas, in the case of a quadratic equation where the power is 2, it gives us two roots. The difference between cubic and quadratic equations is really important to solve these types of questions.