Polynomials question- help please!!! (1 Viewer)

[NizDiz]

Hey peeps, just a couple of question I'm unsure with.

1. A monic cubic polynomial when divided by x^2 +4 leaves a remainder of x + 8 and when divided by x leaves a remainder of -4. Find the polynomial in the form ax^3 + bx^2 + cx +d =0 (I know that a=1 ;given and d =-4) (Answer: x^3 - 3x^2 +5x -4)

2. Two of the roots of the equation x^3 + ax^2 + bx =0 are reciprocals of each other (a,b are both real)

(a) Show that a = b - 1/b

(b) Show that the two roots, which are reciprocal, will be real if -1/2 </= b </= 1/2

(Note; I've already proved that -b is third root, so you can just use it)

All help would be greatly appreciated. Thank you

 

[Makematics]

just wondering, for the second question, is it meant to be x^3 + ax^2 + bx+c =0 or is it right as it is written? because if it is already like that, you know 0 is a root by the factor theorem, and you can use the sum of roots taken two at a time to find that b=1, which doesnt sound like what the question wants :/
also, 2(b) doesnt make much sense.

 

[NizDiz]

yeh my bad for 2 (b), b it meant to be greater than or equal to -1/2 or less than or equal to 1/2. question 2 is done right, still don't get how u did it though... How do u do 1?