1. ## Potential Rational Zeros

List the potential rational zeros of the polynomial function below. Do NOT attempt to find the zeros.

f(x) = -6x^3 - x^2 + x + 10

How do I make a list of such zeros?

2. Use the Rational Root Test:
The Rational Roots Test

3. Originally Posted by magentarita
List the potential rational zeros of the polynomial function below. Do NOT attempt to find the zeros.

f(x) = -6x^3 - x^2 + x + 10

How do I make a list of such zeros?
Use the Rational Zeros Theorem:

P: List all possible factors of leading coefficient (-6): -6, -3, -2, -1, 1, 2, 3, 6
Q: List all possible factors of lone constant (10): -10, -5, -2, -1, 1, 2, 5, 10

List all numbers $\displaystyle \frac{P}{Q}$, all possible solutions lie in the set

I would find it much easier to do some factoring...

f(x)=x(-6x^2-x+1)+10

$\displaystyle f(x)=x(2x+1)(-3x+1)+10$

To have $\displaystyle f(x)=0$, we need $\displaystyle x(2x+1)(1-3x)=-10$

Let's try x = 1, this gives us -6
x = 2 gives us -50

Our answer must lie between 1 and 2, so what possible solutions, from the Rational Zeros theorem are between 1 and 2?

$\displaystyle \frac{5}{3}, \frac{6}{5}, \frac{3}{2}$

Can you finish this out now?

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Is LaTeX not working for some reason?

4. ## Fantastic!

I thank you both, especially Colby for your detailed explanation.

5. Originally Posted by magentarita
List the potential rational zeros of the polynomial function below. Do NOT attempt to find the zeros.

f(x) = -6x^3 - x^2 + x + 10

How do I make a list of such zeros?
The rational root theorem tells you that any rational root of a polynomial with integer coeficients is the ratio of a divisor (positive or negative) of the constant term to a divisor (positive or negative) of the coefficient of the highest order term.

The divisors of -6 are +/- 1, +/- 2, +/- 3, +/- 6, and of 10 are +/- 1, +/-2, +/- 5 +/- 10.

RonL

6. ## Great math notes.

Your replies are helping me more and more each day.