Prove that `|1alphaalpha^2alphaalpha^2 1alpha^2 1alpha|=-(1-alpha^3)^2dot`

Prove that |{: (1,alpha,alpha^2), (alpha,alpha^2, 1),(alpha^2, 1,alpha) :}|=-(1-alpha^3)^2

Find the coefficient of alpha^(6) in the product (1+alpha+alpha^(2))(1+alpha+alpha^(2))(1+alpha+alpha^(2)+alpha^(3)) (1+alpha)(1+alpha)(1+alpha) .

Find the coefficient of alpha^(6) in the product (1+alpha+alpha^(2))(1+alpha+alpha^(2))(1+alpha+alpha^(2)+alpha^(3)) (1+alpha)(1+alpha)(1+alpha) .

Find the coefficient of alpha^(6) in the product (1+alpha+alpha^(2))(1+alpha+alpha^(2))(1+alpha+alpha^(2)+alpha^(3)) (1+alpha)(1+alpha)(1+alpha) .

Find the coefficient of alpha^(6) in the product (1+alpha+alpha^(2))(1+alpha+alpha^(2))(1+alpha+alpha^(2)+alpha^(3)) (1+alpha)(1+alpha)(1+alpha) .

Find the coefficient of alpha^(6) in the product (1+alpha+alpha^(2))(1+alpha+alpha^(2))(1+alpha+alpha^(2)+alpha^(3)) (1+alpha)(1+alpha)(1+alpha) .

If sum_(n=1)^nalpha_n=a n^2+b n ,w h e r ea ,b are constants and alpha_1,alpha_2,alpha_3 in {1,2,3,......,9}a n d25alpha_1,37alpha_2,49alpha_3 be three digit number, then prove that |alpha_1alpha_2alpha_3 5 7 9 25alpha_1 37alpha_2 49alpha_3|=0

If sum_(n=1)^nalpha_n=a n^2+b n ,w h e r ea ,b are constants and alpha_1,alpha_2,alpha_3 in {1,2,3,......,9}a n d25alpha_1,37alpha_2,49alpha_3 be three digit number, then prove that |alpha_1alpha_2alpha_3 5 7 9 25alpha_1 37alpha_2 49alpha_3|=0

If sum_(n=1)^nalpha_n=a n^2+b n ,w h e r ea ,b are constants and alpha_1,alpha_2,alpha_3 in {1,2,3,......,9}a n d25alpha_1,37alpha_2,49alpha_3 be three digit number, then prove that |alpha_1alpha_2alpha_3 5 7 9 25alpha_1 37alpha_2 49alpha_3|=0

If the roots of equation x^3+a x^2+b=0a r ealpha_1,alpha_2 and alpha_3(a ,b!=0) , then find the equation whose roots are (alpha_1alpha_2+alpha_2alpha_3)/(alpha_1alpha_2alpha_3),(alpha_2alpha_3+alpha_3alpha_1)/(alpha_1alpha_2alpha_3),(alpha_1alpha_3+alpha_1alpha_2)/(alpha_1alpha_2alpha_3)