solve 3x^3+x^2-8x+4>0. show all the algebraic calculations. (5 marks)

Answer 1

look for low-hanging fruit first

when x=1, y=3+1-8+4 = 0, so we have
(x-1)(3x^2+4x-4)
That factors into
(x-1)(3x-2)(x+2)

Now, from our general knowledge of cubics, we know that y<0 to the left of the smallest root. Then y alternates signs between roots, so we have y>0 for xin the intervals

(-2,2/3) and (1,∞)