How do you factor the expression ${x^4} - 13{x^2} + 36$?

Hint:Here we have to factorize the polynomial ${x^4} - 13{x^2} + 36$. Factorization is defined as the process in which we break a number or a polynomial into a product of many factors of other polynomials, which when multiplied gives the original number. We will factorize the polynomial by splitting the term and using the difference of squares identity i.e., ${a^2} - {b^2} = (a - b)(a + b)$.

Complete step by step answer:
Factorization is defined as the process of expressing or decomposing a number or an algebraic expression as a product of its factors.In the question we have to factorize the polynomial ${x^4} - 13{x^2} + 36$. Polynomial can be defined as the expression consisting of coefficients and variables which are also known as intermediates. These are generally a sum or difference of variables and exponents. Each part of the polynomial is known as “term”.

We have ${x^4} - 13{x^2} + 36$. Here we can see that we can split $13{x^2}$ into $9{x^2} + 4{x^2}$ as $36 = 4 \times 9$. So, the above equation can be written as,
$ \Rightarrow {x^4} - 9{x^2} - 4{x^2} + 36$
Taking ${x^2}$ and $4$ as common factor in the above expression we get,
$ \Rightarrow {x^4} - 9{x^2} - 4{x^2} + 36 = {x^2}({x^2} - 9) - 4({x^2} - 9)$
The above expression can be written as
$ \Rightarrow ({x^2} - 4)({x^2} - 9)$
Now using the identity ${a^2} - {b^2} = (a - b)(a + b)$. We get,
$ \therefore ({x^2} - 4)({x^2} - 9) = (x - 2)(x + 2)(x - 3)(x + 3)$
Hence, $(x - 2)(x + 2)(x - 3)(x + 3)$ is the factor of ${x^4} - 13{x^2} + 36$.

Note:In order to solve these types of problems in which we have to factorize the polynomial using splitting of terms we can only split the term in which the $x$ term is the sum of two factors and product equal to the constant term. In factorization there is an important term ‘Factor’. Factor is defined as the numbers, algebraic variables or an algebraic expression which divides the number or an algebraic expression without leaving any remainder.