Question

Factorise.
(i)\[{{\rm{a}}^4}{\rm{ - }}{{\rm{b}}^4}\]
(ii)\[{{\rm{p}}^4}{\rm{ - 81}}\]

Answer 1

Hint: Here, we have to use the concept of the factorization. Factorization is the process in which a number is written in the forms of its small factors which on multiplication give the original number. So by using the basic properties of \[{{\rm{x}}^2}{\rm{ - }}{{\rm{y}}^2}\] we will be able to factorize both of the given functions.

Formula used:
\[{{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ = (x + y)(x - y)}}\]

Complete step-by-step answer:
(i)Given function is \[{{\rm{a}}^4}{\rm{ - }}{{\rm{b}}^4}\]
Factors are the smallest numbers or terms with which the given number is divisible and their multiplication will give the original number.
We can also write the given equation \[{{\rm{a}}^4}{\rm{ - }}{{\rm{b}}^4}\]as \[{{\rm{(}}{{\rm{a}}^2}{\rm{)}}^2}{\rm{ - (}}{{\rm{b}}^2}{)^2}\]
So, using the formula of \[{{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ = (x + y)(x - y)}}\] where \[{\rm{x = }}{{\rm{a}}^2}\] and \[{\rm{y = }}{{\rm{b}}^2}\]
Therefore, the equation becomes \[{{\rm{a}}^4}{\rm{ - }}{{\rm{b}}^4}{\rm{ = (}}{{\rm{a}}^2} + {{\rm{b}}^2})({{\rm{a}}^2}{\rm{ - }}{{\rm{b}}^2})\]
We can further factorize this equation by expanding the term\[({{\rm{a}}^2}{\rm{ - }}{{\rm{b}}^2})\] in the above equation using the same formula of \[{{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ = (x + y)(x - y)}}\]where\[{\rm{x = a,y = b}}\], we get
\[{{\rm{a}}^4}{\rm{ - }}{{\rm{b}}^4}{\rm{ = (}}{{\rm{a}}^2} + {{\rm{b}}^2})({\rm{a + b}})({\rm{a - b}})\]
Hence, \[{{\rm{a}}^4}{\rm{ - }}{{\rm{b}}^4}\]is factorized

(ii)Given function is \[{{\rm{p}}^4}{\rm{ - 81}}\]
We can also write the given equation \[{{\rm{p}}^4}{\rm{ - 81}}\]as \[{{\rm{(}}{{\rm{p}}^2}{\rm{)}}^2}{\rm{ - (9}}{)^2}\]
So, by using the formula of \[{{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ = (x + y)(x - y)}}\] where \[{\rm{x = }}{{\rm{p}}^2}\] and \[{\rm{y = }}9\]
Therefore, the equation becomes \[{{\rm{p}}^4}{\rm{ - 81 = (}}{{\rm{p}}^2}{\rm{ + 9)(}}{{\rm{p}}^2}{\rm{ - 9)}}\]
We can further factorize this equation by expanding the term \[{\rm{(}}{{\rm{p}}^2}{\rm{ - 9)}}\] in the above equation
We can also write \[{\rm{(}}{{\rm{p}}^2}{\rm{ - 9)}}\]as \[{\rm{(}}{{\rm{p}}^2}{\rm{ - }}{{\rm{3}}^2}{\rm{)}}\] using the same formula of\[{{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ = (x + y)(x - y)}}\]where \[{\rm{x = p,y = 3}}\], we get
\[{{\rm{p}}^4}{\rm{ - 81 = (}}{{\rm{p}}^2}{\rm{ + 9)(p + 3)(p - 3)}}\]
Hence, \[{{\rm{p}}^4}{\rm{ - 81}}\]is factorized.

Note: Algebraic identities in maths refer to an equation that is always true regardless of the values assigned to the variables. Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation. An algebraic identity is an equality that holds for any values of its variables. We have to remember that mathematics identity can be solved by using this approach and also we have to remember the basic mathematics identity. It is very important that we learn about algebraic identities in maths.
\[{\left( {{\rm{a }} + {\rm{ b}}} \right)^2}\; = {\rm{ }}{{\rm{a}}^2}\; + {\rm{ }}2{\rm{ab }} + {\rm{ }}{{\rm{b}}^2}\]
\[{\left( {{\rm{a - b}}} \right)^2}\; = {\rm{ }}{{\rm{a}}^2}\; - {\rm{ }}2{\rm{ab }} + {\rm{ }}{{\rm{b}}^2}\]
\[{{\rm{a}}^2}{\rm{ - }}{{\rm{b}}^2} = {\rm{(a - b)}}({\rm{a + b)}}\]