Question

How do you factor: $ \left( {{x^3} + 13{x^2} + 42x} \right) $

Answer 1

Hint: Given polynomial is of degree $ 3 $ . Polynomials of degree $ 3 $ are known as cubic polynomials. The given cubic polynomial can be converted into a quadratic polynomial by factoring out the x that is common in all the terms of the polynomial. So, the given polynomial can be converted into a cubic polynomial by the above mentioned process. Quadratic polynomials can be factored by the help of splitting the middle term method. In this method, the middle term is split into two terms in such a way that the polynomial remains unchanged.

Complete step-by-step answer:
For factorising the given cubic polynomial $ \left( {{x^3} + 13{x^2} + 42x} \right) $ , we factor out the x common in all the terms of the polynomial.
So, $ \left( {{x^3} + 13{x^2} + 42x} \right) = x\left( {{x^2} + 13x + 42} \right) $
Now, we have to factorise the quadratic polynomial expression thus obtained. We can use splitting the middle term method in which the middle term is split into two terms such that the sum of the terms gives us the original middle term and product of the terms gives us the product of the constant term and coefficient of $ {x^2} $ .
So, $ \left( {{x^2} + 13x + 42} \right) $
 $ = $ $ {x^2} + \left( {6 + 7} \right)x + 42 $
We split the middle term $ 13x $ into two terms $ 6x $ and $ 7x $ since the product of these terms, $ 42{x^2} $ is equal to the product of the constant term and coefficient of $ {x^2} $ and sum of these terms gives us the original middle term, $ 13x $ .
 $ = $ $ {x^2} + 6x + 7x + 42 $
Taking x common from the first two terms and $ 7 $ common from the last two terms. We get,
 $ = $ $ x\left( {x + 6} \right) + 7\left( {x + 6} \right) $
 $ = $ $ \left( {x + 6} \right)\left( {x + 7} \right) $
So, the factored form of the quadratic polynomial $ \left( {{x^3} + 13{x^2} + 42x} \right) $ is $ x\left( {x + 6} \right)\left( {x + 7} \right) $ .
So, the correct answer is “ $ x\left( {x + 6} \right)\left( {x + 7} \right) $ ”.

Note: Splitting of the middle term can be a tedious process at times when the product of the constant term and coefficient of $ {x^2} $ is a large number with a large number of divisors. Special care should be taken in such cases. Similar to quadratic polynomials, quadratic solutions can also be solved using factorisation method. Besides factorisation, there are various methods to solve quadratic equations such as completing the square method and using the Quadratic formula.