Question

Factorize each of the following algebraic expression:
(a² − 5a)² − 36

Answer 1

$$\begin{gathered} (a^2-5a{)}^2-36 \\ =(a^2-5a{)}^2-6^2 \\ =\left[\right(a^2-5a)-6]\left[\right(a^2-5a)+6] \\ =(a^2-5a-6)(a^2-5a+6) \\ Inordertofactorise{a}^2-5a-6,wewillfindtwonumberspandqsuchthatp+q=-5andpq=-6 \\ Now, \\ (-6)+1=-5 \\ and \\ (-6)\times 1=-6 \\ Splittingthemiddleterm-5inthegivenquadraticas-6a+a,weget: \\ {a}^2-5a-6=a^2-6a+a-6 \\ =(a^2-6a)+(a-6) \\ =a(a-6)+(a-6) \\ =(a+1)(a-6) \\ Now, \\ Inordertofactorise{a}^2-5a+6,wewillfindtwonumberspandqsuchthatp+q=-5andpq=6 \\ Clearly, \\ (-2)+(-3)=-5 \\ and \\ (-2)\times (-3)=6 \\ Splittingthemiddleterm-5inthegivenquadraticas-2a-3a,weget: \\ {a}^2-5a+6=a^2-2a-3a+6 \\ =(a^2-2a)-(3a-6) \\ =a(a-2)-3(a-2) \\ =(a-3)(a-2) \\ \therefore (a^2-5a-6)(a^2-5a+6)=(a-6)(a+1)(a-3)(a-2) \\ =(a+1)(a-2)(a-3)(a-6) \end{gathered}$$