How do you factor $1 - 2.25{x^8}$ ?

Hint:
In this question we have to factorise the polynomial, and we will do this by using algebraic identity $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$ and then further simplification of the given polynomial, we will get the required factorised term.

Complete step by step solution:
Given polynomial is $1 - 2.25{x^8}$,
Here factorising the terms, we get,
$2.25$can be written as $2.25 = \dfrac{{225}}{{100}} = \dfrac{9}{4}$ ,
Now the given polynomial can be rewritten as ,
$ \Rightarrow 1 - \dfrac{9}{4}{x^8}$,
Now $\dfrac{9}{4} = \dfrac{{3 \times 3}}{{2 \times 2}} = {\left( {\dfrac{3}{2}} \right)^2}$and 1 can be written as $1 = 1 \times 1 = {1^2}$, and ${x^8} = {x^4} \times {x^4} = {\left( {{x^4}} \right)^2}$,
Now using the algebraic identity, ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$, here $a = 1$and $b = 1.5{x^4}$now substituting the values in the identity,
$ \Rightarrow 1 - 2.25{x^8} = {\left( 1 \right)^2} - {\left( {\dfrac{3}{2}{x^4}} \right)^2}$,
Now using the identity, we get,
$ \Rightarrow 1 - 2.25{x^8} = \left( {1 - \dfrac{3}{2}{x^4}} \right)\left( {1 + \dfrac{3}{2}{x^4}} \right)$,
Now simplifying by taking L.C.M, we get,
$ \Rightarrow 1 - 2.25{x^8} = \left( {\dfrac{{2 - 3{x^4}}}{2}} \right)\left( {\dfrac{{2 + 3{x^4}}}{2}} \right)$,
Now again simplifying we get,
$ \Rightarrow 1 - 2.25{x^8} = \dfrac{1}{4}\left( {2 - 3{x^4}} \right)\left( {2 + 3{x^4}} \right)$,
So, by factorising the given polynomial we get $\dfrac{1}{4}\left( {2 - 3{x^4}} \right)\left( {2 + 3{x^4}} \right)$.

$\therefore $ The factorising term when the given polynomial $1 - 2.25{x^8}$ is factorised will be equal to $\dfrac{1}{4}\left( {2 - 3{x^4}} \right)\left( {2 + 3{x^4}} \right)$.

Note:
Factorization is a process which is necessary to simplify the algebraic expressions and is used to solve the higher degree equations. It is the inverse procedure of the multiplication of the polynomials. The algebraic expression is said to be in a factored form only when the whole expression is an indicated product. Factorisation of the algebraic expression can also be done in other methods such as, factoring by grouping and factoring using identities. Algebraic identities are algebraic equations which are always true for every value of variables in them. In an algebraic identity, the left-side of the equation is equal to the right-side of the equation, some of the algebraic identities that are commonly used are:
1) ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$,
2) ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$,
3) $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$,
4) ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$,
5) ${a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$.