1. List all possible rational zeros of f(x)=3x^4-11x^3-x^2+19x+6

To find the possible rational zeros of the polynomial function \( f(x) = 3x^4 - 11x^3 - x^2 + 19x + 6 \), we can use the Rational Root Theorem. This theorem states that any rational zero, expressed as \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient.<br /><br />For the polynomial \( f(x) = 3x^4 - 11x^3 - x19x + 6 \):<br /><br />1. The constant term is \( 6 \).<br />2. The leading coefficient is \( 3 \).<br /><br />First, list the factors of the constant term \( 6 \):<br />\[ \pm 1, \pm 2, \pm 3, \pm 6 \]<br /><br />Next, list the factors of the leading coefficient \( 3 \):<br />\[ \pm 1, \pm 3 \]<br /><br />Now, form all possible fractions \( \frac{p}{q} \) where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient:<br />\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac}{ \pm \frac{2}{3}, \pm \frac{3}{3} = \pm 1, \pm \frac{6}{3} = \pm 2 \]<br /><br />So, the list of all possible rational zeros is:<br />\[ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3} \]<br /><br />Therefore, the possible rational zeros of the polynomial \( f(x) = 3x^4 - 11x^3 - x^2 + 19x + 6 \) are:<br />\[ \boxed{\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm2}{3}} \]