Question

How to factorize the equation $ {x^4} - 5{x^2} + 4 $ ?

Answer 1

Hint: For solving this equation we should know about the factorization. The process of writing a number as a product of several factors. When factors are multiplied together then they form the original number as in the beginning.

Complete step by step solution:
The most common method of factorization is to completely factor the number into its positive prime factors. The number which has only positive factor1 and the number itself, is termed as the prime number.
The factors of any given equation may be an integer, an algebraic expression or a variable or itself.
For example, 2 and 4 are the factors of 8.
There are six methods to solve factorization. These are Greatest Common Factor, Grouping Method, Sum or difference in two cubes, Difference in two squares method, General trinomials and Trinomial method.
Now we will solve the following equation by the use of factorization method.
Given that the equation is
 $ {x^4} - 5{x^2} + 4 $
By splitting the middle term, we get
 $ {x^4} - 4{x^2} - {x^2} + 4 $ .
On taking common factors we get
 $ {x^2}\left( {{x^2} - 4} \right) - 1\left( {{x^2} - 4} \right) $
On simplifying we get
 $ \left( {{x^2} - \,\,4} \right)\,\left( {{x^2}\, - \,1} \right) $
 $ \left( {{x^2} - {2^2}} \right)\,\left( {{x^2} - {1^2}} \right) $
We know that
 $ {x^2} - {b^2} = \left( {x + b} \right)\,\left( {x - b} \right) $
On using above formula we get
 $ \left( {x + 2} \right)\,\left( {x - 2} \right)\,\left( {x + 1} \right)\,\left( {x - 1} \right) $
Now on putting this equation equal to zero, we will get the factors of x.
 $ \left( {x + 2} \right)\,\left( {x - 2} \right)\,\left( {x + 1} \right)\,\left( {x - 1} \right)\,\, = \,\,0 $
Hence, the values of x are 2, -2, 1 and -1.
 $ \left( {x + 2} \right)\,\left( {x - 2} \right)\,\left( {x + 1} \right)\,\left( {x - 1} \right) $ is the required factorization.
So, the correct answer is “ $ \left( {x + 2} \right)\,\left( {x - 2} \right)\,\left( {x + 1} \right)\,\left( {x - 1} \right) $ ”.

Note: Factorization is an important process, which helps us understand more about the equations. Through factorization, usually we rewrite the polynomials in its simpler form, and when we use factorization’s principles in any equation, we find a lot of useful information about it. The ancient Greek Mathematicians were the first one who considered the factorization in the case of integers.