Question

Divide $44\left( {{x^4} - 5{x^3} - 24{x^2}} \right)$ by $11x\left( {x - 8} \right)$.

Answer 1

Hint: We can use the long division method or factorization method to solve this type of sum.
Here we use the Factorization method, which is the process of reducing the bracket of a quadratic equation, instead of expanding the bracket and converting the equation to a product of factors that cannot be further.
While factoring a polynomial, these two points in mind:
Finding factors of the polynomial is just like doing any simple division, the only thing to be kept in mind is the accuracy of variables and coefficients.
Factorization by division method is the conventional method of finding factors of a polynomial equation.
Using the above rule we divide the polynomial.

Complete step-by-step solution:
It is given that the equation we can write It as,
$ \Rightarrow \dfrac{{44\left( {{x^4} - 5{x^3} - 24{x^2}} \right)}}{{11x\left( {x - 8} \right)}}$ ……………….... (1)
Here, the given form $44\left( {{x^4} - 5{x^3} - 24{x^2}} \right)$ is considered as a polynomial to break the form.
To factorize, the given polynomial first take ${x^2}$ commonly from each term,
$ \Rightarrow 44{x^2}\left( {{x^2} - 5x - 24} \right)$
We have to break $ - 24$ by finding its divisors, its sum must be $ - 5$, the middle term.
We can write -5 as (3 – 8),
$ \Rightarrow 44{x^2}\left[ {{x^2} + \left( {3 - 8} \right)x - 24} \right]$
Hence the given polynomial will be written as
$ \Rightarrow 44{x^2}\left( {{x^2} + 3x - 8x - 24} \right)$
Taking the common terms and we can write it as,
$ \Rightarrow 44{x^2}\left[ {x\left( {x + 3} \right) - 8\left( {x + 3} \right)} \right]$
Here we can take $\left( {x + 3} \right)$ common,
$ \Rightarrow 44{x^2}\left( {x + 3} \right)\left( {x - 8} \right)$
Now, the polynomial is broken into two factors, from now, the consideration of polynomials will fall.
$ \Rightarrow 44\left( {{x^4} - 5{x^3} - 24{x^2}} \right) = 44{x^2}\left( {x + 3} \right)\left( {x - 8} \right)$
So, substitute the factorized part in equation (1),
$ \Rightarrow \dfrac{{44{x^2}\left( {x + 3} \right)\left( {x - 8} \right)}}{{11x\left( {x - 8} \right)}}$
Cancel out the common factors,
$\therefore 4x\left( {x + 3} \right)$

Hence, the required answer is $4x\left( {x + 3} \right)$.

Note: The alternative method is the normal division method. It is very easy to understand and solving will be easy, but we use factorization, when the factors can be found it is also very easy to solve.
The factorization is the reverse function of multiplication. A form of disintegration, factorization entails the gradual breakdown of a polynomial into its factors.