We are given the next polynomial, and are asked to factor it:
[tex]25a^2\text{ - }10a+1[/tex]We are going to use the perfect square formula, which states that:
[tex]b^2\text{ - }2bc+c^2=(b\text{ -}c)^2[/tex]Therefore, we need to rewrite our polynomial. We start by factoring the number 25 and applying the exponent rule, as follows:
[tex]25a^2\text{ -->}5^2a^2\text{ --> }(5a)^2[/tex]Now, we factor the second term, as follows:
[tex]\text{ -}10a\text{ --> -}2(5a)\text{ --> We add a 1, which doesn't change anything --> -}2(5a)(1)[/tex]Finally, we rewrite our last term as:
[tex]1\text{ --> }1^2[/tex]With these, we have that our initial polynomial is:
[tex]25a^2\text{ - }10a+1\text{ --> }(5a)^2\text{ - }2(5a)(1)+1^2[/tex]And we see that we already have it in the form that we need to apply the perfect square formula, in which 5a is b and 1 is c, so we apply it just as follows:
[tex](5a)^2\text{ - }2(5a)(1)+1^2\text{ --> }(5a\text{ - }1)^2[/tex]And therefore, (5a - 1)² is the answer
