How do you simplify #(p^3-1)/(5-10p+5p^2)#?

Question

Eva Abramson · Accepted Answer

#(p^3-1)/(5-10p+5p^2)=(p^2+p+1)/(5(p-1))# or #(p^2+p+1)/(5p-5)# To simplify #(p^3-1)/(5-10p+5p^2)#, we should first factorize numerator and denomiantor. #p^3-1=p^3-p^2+p^2-p+p-1# = #p^2(p-1)+p(p-1)+1(p-1)# = #(p^2+p+1)(p-1)# and #5-10+5p^2=5(p^2-2p+1)# = #5(p^2-p-p+1)=5(p(p-1)-1(p-1))=5(p-1)(p-1)# Hence #(p^3-1)/(5-10p+5p^2)# = #((p^2+p+1)(p-1))/(5(p-1)(p-1))# = #((p^2+p+1)cancel((p-1)))/(5(p-1)cancel((p-1)))# = #(p^2+p+1)/(5(p-1))# or #(p^2+p+1)/(5p-5)#

Nathan Axel · Answer

undefined To simplify the expression (p^3-1)/(5-10p+5p^2), we can factor the numerator and denominator separately. The numerator can be factored as a difference of cubes, and the denominator can be factored as a trinomial. Factoring the numerator, we get (p-1)(p^2+p+1). Factoring the denominator, we get 5(1-2p+p^2), which simplifies to 5(p-1)^2. Therefore, the expression simplifies to (p-1)(p^2+p+1)/(5(p-1)^2).