# Sum & product of roots of a polynomial

#### xc630

##### Junior Member
Hi I have to find the sum and product of the roots of 3x^3+6x^2-2x+9. I'm having trouble finding the roots. I tried synthetic division, testing +/- 1,3,and 9 but none were zeroes. I don't know if I did this wrong. I also tried factoring but didn't get very far. I would appreciate any help. Thanks.

#### soroban

##### Elite Member
Hello, xc630!

I believe they expect you to know a theorem for this problem.

I have to find the sum and product of the roots of 3x<sup>3</sup> + 6x<sup>2</sup> - 2x + 9 .= .0
.
In baby-talk, the theorem goes like this:

Given: .x<sup>3</sup> + ax<sup>2</sup> + bx + c .= .0 . . . Note: the leading coefficient is 1.

If the three roots are p, q, r, then:
. . . . .p + q + r .= .-a . . . the sum of the roots is the negative of a.
. . pq + qr + pr .= .b
. . . . . . . . .pqr .= .-c . . . the product of the roots is the negative of c.

Your equation is: .3x<sup>3</sup> + 6x<sup>2</sup> - 2x + 0 .= .0

Divide by 3: . x<sup>3</sup> + 2x<sup>2</sup> - (2/3)x + 3 .= .0

. . . . . .Sum of roots: . -2
. . . Product of roots: . -3

#### xc630

##### Junior Member
but how do I find the 3 roots?

#### stapel

##### Super Moderator
Staff member
Review soroban's reply. You were asked for the sum and product; not the actual roots. He gave you the formula for the sum and product of the roots. Use the formula he gave you to obtain the answers he gave you.

Eliz.

#### xc630

##### Junior Member
I am confused. Using his formula, what would be p,q , and r in my case?

#### stapel

##### Super Moderator
Staff member
Why do you need to find p, q, or r? Did the question ask you for that, or did it ask you for the sum and product of the roots? Did the previous post give you the formula (that your question assumed you'd learned) for the obtaining the sum and the product? And did the previous post also give you the answer to your question?

I'm sorry, but you've been provided the fully-worked solution. I'm not understanding what you still need.

Eliz.

#### xc630

##### Junior Member
nvr mind. thank you.