Question

How do you factor completely \[{a^4} - {b^4}\] ?

Answer 1

Hint: In this question we used the formula for factor; the power of ‘a’ is two minus the power of ‘b’ is two. The expression of the power of ‘a’ is two minus power of ‘b’ is two is called the difference of squares. And this formula is expressed as below.
\[ \left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)\]
This is the formula for the difference of squares.

Complete step by step answer:
We know that, we find the factor of \[\left( {{a^2} - {b^2}} \right)\]form type.
First, we want to calculate the factor of the above formula.
Then,
\[{a^2} - {b^2}\]
In the above expression, we can add and subtract \[ab\]. By adding and subtracting \[ab\], there is no effect in the above expression.
Then,
The above expression is written as below.
\[ {a^2} - {b^2} + ab - ab\]
This is written as below form.
\[{a^2} - ab + ab - {b^2}\]
Then we take the common, in the first two we take \[a\] as common and in the last two we take \[b\] as common.
Then the above expression is written as below.
\[ a\left( {a - b} \right) + b\left( {a - b} \right)\]
By solving the above expression, the result would be as below.
\[\left( {a - b} \right)\left( {a + b} \right)\]
Then, we prove that \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\].
Let’s come to the question, in the question the data is given below.
\[ {a^4} - {b^4}\]
We can write \[{a^4}\]in the form of \[{\left( {{a^2}} \right)^2}\] and also \[{b^4}\] in the form of \[{\left( {{b^2}} \right)^2}\].
Then the above expression is written as.
\[ {\left( {{a^2}} \right)^2} - {\left( {{b^2}} \right)^2}\]
We know that \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\]. Then the above expression is written as.
\[ {\left( {{a^2}} \right)^2} - {\left( {{b^2}} \right)^2} = \left( {{a^2} - {b^2}} \right)\left( {{a^2} + {b^2}} \right)\]
Now we again apply the difference of the square formula.
Then,
\[ = \left( {a - b} \right)\left( {a + b} \right)\left( {{a^2} + {b^2}} \right)\]

Therefore, the factor of \[{a^4} - {b^4}\] are \[\left( {a - b} \right)\left( {a + b} \right)\left( {{a^2} + {b^2}} \right)\].

Note:
In this question, we find the factor of \[{a^2} - {b^2}\]. The formula for finding the factor of \[{a^2} - {b^2}\] type is derived by adding and subtracting the \[ab\] term. If the number which comes in place of \[a\] and \[b\] is a perfect square. Then the factor can easily find out.