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 Root Test:
The Rational Roots Test
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?
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