Question

Factorize
${p^4} - 81$

Answer 1

Hint: This question is nothing but just an application of a single formula of algebra. It would become very easy if you identify the number given in the question as a root of which number. Factorization is a process in which we find the roots of a given equation.

Complete step-by-step answer:
In the given question,
$ \Rightarrow {p^4} - 81$
We can also write $81 = {3^4}$
$ \Rightarrow {p^4} - {3^4}$
$ \Rightarrow {\left( {{p^2}} \right)^2} - {\left( {{3^2}} \right)^2}$
Using formula, ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
Where $a = {p^{2\,}}\,,\,\,b = {3^2}$
$ \Rightarrow \left( {{p^2} + {3^2}} \right)\left( {{p^2} - {3^2}} \right)$
Again, using the formula ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
Where $a = p\,\,and\,\,b = 3$
$ \Rightarrow \left( {{p^2} + {3^2}} \right)\left( {p + 3} \right)\left( {p - 3} \right)$
Therefore,
$ \Rightarrow \left( {{p^2} + 9} \right)\left( {p + 3} \right)\left( {p - 3} \right)$
Hence, this is our required factorized value.

Note: While doing factorization one must know all the formulas in algebra regarding square and cube of numbers. Factoring an algebraic expression means writing the expression as a product of factors.
To verify whether the factors are correct or not, multiply them and check if you get the original algebraic expression. An algebraic expression can be factored using the common factor method, regrouping like terms together, and also by using algebraic identities.