Which of the following is the complete list of roots for the polynomial function f(x) = (x2 + 2x - 15)(x2 + 8x + 17)? a) -5,3 b) -5,3,-4,+i,-4,-i c) -5,3,-4,+i,4+i d) -4+i , -4 -i

The first one factors easily.  The first quadratic, I mean.  The 2 numbers that add up to +2 and at the same time multiply to -15 and 5 and -3.  So those are 2 of the 4 roots we have.  The second quadratic does not factor so nicely.  You need to put that into the quadratic formula to solve.    which simplifies to  .  That gives us a negative radicand and that's a problem.  .  Since -1 is equal to i^2, we can rewrite to begin dealing with the negative properly.  .  Replacing -1 with i^2 gives us  .  i^2 has a perfect root of i in it, and 4 has a perfect square of 2 in it, so we simplify more to  .  The 2 in the denominator reduces with the numerator to give us a final 2 roots that are  x = -4 + i,  and x = -4 - i.  Taking all those roots together, we find that the solution to our problem is choice B (although I believe you put some extra commas in there on accident).

Given polynomial function f(x) = (x^2 + 2x - 15)(x^2 + 8x + 17).

In order to find all roots of the given polynomial function, we need to set each of above factor equal to 0.

Therefore,

x^2 + 2x - 15 =0 and x^2 + 8x + 17 =0.

Let us solve first quadratic now.

x^2 + 2x - 15 =0

Factoring out quadratic

(x+5)(x-3) =0

x+5=0 or x-3=0

x=-5 or x=3.

Let us solve second quadratic equation now

x^2 + 8x + 17 =0.

Applying quadratic formula,

Therefore, the list of roots for the polynomial function is:

So, the correct option is B) -5, 3, -4+i, -4-i.

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