# Thread: Solving for x: polynomial equation

1. ## Solving for x: polynomial equation

(I really hope this is in the correct place.)

The equation is $\displaystyle 3x^4 + 8x^3 - 30x^2 -72x +27$ I need to find the four roots but I only managed three, and one was by looking and guessing. In an exam, I won't have time to start guessing so I was wondering if there was a more methodic, secure way to solve this.

I guessed (using the calculator) that 3 was one root. Through that, I guessed -3. Then I divided the equation by $\displaystyle (x^2 - 9)$ and got $\displaystyle 3x^2 + 8x - 3$ which, in turn, gave me $\displaystyle x=\frac{1}{3}$ and $\displaystyle x=-3$. But that's what I don't understand, I divided the original equation by $\displaystyle (x+3)$, how am I still getting that as a result? And since the highest power is 4, I assumed that I would find four roots but I've only found three.

Also, once I have all four answers, the question stated that I need to write them out in order, from the most amount of times that the root appears in the equation. I don't exactly understand what this means. I would post the original question, but it's not in English and therefore wouldn't really be of any value. I translated as best as I could.

Thank you.

2. Originally Posted by dantap
(I really hope this is in the correct place.)

The equation is $\displaystyle 3x^4 + 8x^3 - 30x^2 -72x +27$ I need to find the four roots but I only managed three, and one was by looking and guessing. In an exam, I won't have time to start guessing so I was wondering if there was a more methodic, secure way to solve this.

I guessed (using the calculator) that 3 was one root. Through that, I guessed -3. Then I divided the equation by $\displaystyle (x^2 - 9)$ and got $\displaystyle 3x^2 + 8x - 3$ which, in turn, gave me $\displaystyle x=\frac{1}{3}$ and $\displaystyle x=-3$. But that's what I don't understand, I divided the original equation by $\displaystyle (x+3)$, how am I still getting that as a result? And since the highest power is 4, I assumed that I would find four roots but I've only found three.

Also, once I have all four answers, the question stated that I need to write them out in order, from the most amount of times that the root appears in the equation. I don't exactly understand what this means. I would post the original question, but it's not in English and therefore wouldn't really be of any value. I translated as best as I could.

Thank you.

Assuming you did the calculations correctly it means that -3 is a double root, so the roots are -3, -3, 1/3, 3.

Tonio

3. The easiest way is to check the factors of $\displaystyle 27$. Any linear factors will be of the form $\displaystyle x - a$ where $\displaystyle a$ is a factor of $\displaystyle 27$.

4. You have to guess the first root to find the other.For this root you have to search for the numbers wich divide 27.These are +1,-1,+3,-3,+9,-9.