2009/032) factor f(x) = x^3-3x^2+4

by taking factors of 4 that is 1,-1.2,-2,4,-4 we see that

f(-1) = 0

so (x+1) is a factor

so x^3-3x^2+4
= x^3+x^2-4(x^2-1)
= x^2(x+1)-4(x+1)(x-1)
= (x+1)(x^2-4(x-1)
= (x+1)(x^2-4x+4)
= (x+1)(x-2)^2

2009/031) simplify sin10*sin50*sin60*sin70*sin90

we know sin 90 = 1 and sin 60 = sqrt(3)/2
hence
sin10*sin50*sin60*sin70*sin90 = sqrt(3/2) sin 10 sin 50 sin 70
= sqrt(3)/2 sin 10 sin (90-40) sin (90-20)
= sqrt(3)/2 sin 10 cos 40 cos 20 ( as sin (90-x) = cos x)
= sqrt(3)/2 sin 10 cos 20 cos 40 ..1

Knowing sin 2a =2 sin a cos a and 20 is double of 10 and 40 is double od 20 we proceed

now sin10 cos 20 cos 40 = (cos 10 sin 10 cos 20 cos 40)/cos 10
= (2 cos 10 sin 10 cos 20 cos 40)/(2 cos 10)
= ( sin 20 cos 20 cos 40)/( 2 cos 10
= (2 sin 20 cos 20 cos 40)/( 4 cos 10)
= ( sin 40 cos 40)/( 4 cos 10)
= ( 2 sin 40 cos 40)/( 8 cos 10)
= sin 80 /( 8 cos 10)
= cos 10/(8 cos 10) = 1/8 ...2

from 1 and 2 we get
sin10*sin50*sin60*sin70*sin90 = sqrt(3)/16