Question

Factorize \[{x^4} + \dfrac{4}{{{x^4}}}\]

Answer 1

Hint:We use some basic properties of algebra to solve this question. First, we apply the basic algebraic property that \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\], then we use the property, \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\], then, we will have the required factors. We should also remember the squares of some numbers like here we are required to know that 4 is the square of 2. After applying the properties we get two terms that are in product form, thus they will be factors.

Complete step by step answer:
We are given \[{x^4} + \dfrac{4}{{{x^4}}}\] and we have to factorize it. First, we know 4 is square of 2 and \[{x^4}\] is square of \[{x^2}\] , and also we know the algebraic property that \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] , so applying this property, we get,
\[{\left( {{x^2} + \dfrac{2}{{{x^2}}}} \right)^2} = {x^4} + \dfrac{4}{{{x^4}}} + 4\]
Now, finding the value of \[{x^4} + \dfrac{4}{{{x^4}}}\] from the above equation, by taking 4 to the other side, so, we get,
\[{x^4} + \dfrac{4}{{{x^4}}} = {\left( {{x^2} + \dfrac{2}{{{x^2}}}} \right)^2} - 4\]
Now, we know that 4 is square of 2 so, we write above equation as,
\[{x^4} + \dfrac{4}{{{x^4}}} = {\left( {{x^2} + \dfrac{2}{{{x^2}}}} \right)^2} - {\left( 2 \right)^2}\]

Now, we apply the algebraic property that, \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\] , we get,
\[{x^4} + \dfrac{4}{{{x^4}}} = \left( {{x^2} + \dfrac{2}{{{x^2}}} - 2} \right)\left( {{x^2} + \dfrac{2}{{{x^2}}} + 2} \right)\]
On factoring \[{x^4} + \dfrac{4}{{{x^4}}}\] , we get \[\left( {{x^2} + \dfrac{2}{{{x^2}}} - 2} \right)\left( {{x^2} + \dfrac{2}{{{x^2}}} + 2} \right)\] this implies, \[\left( {{x^2} + \dfrac{2}{{{x^2}}} - 2} \right)\] and \[\left( {{x^2} + \dfrac{2}{{{x^2}}} + 2} \right)\] are factors of \[{x^4} + \dfrac{4}{{{x^4}}}\] as the product of these two terms is \[{x^4} + \dfrac{4}{{{x^4}}}\].

Note:Factoring means dividing the term in such a way that on multiplication of all its terms (that are known as factors) we get the given number. We use some of the basic algebraic properties to solve this question. Keep in mind the algebraic properties like, \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] and \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\] to solve these types of questions. Take care while doing the calculations. Use these properties carefully. You should also keep in mind the squares of numbers.