Questions: Find all the zeros. Write the answer in exact form. p(x)=3 x^4-14 x^3+15 x^2+12 x-20 If there is more than one answer, separate them with commas. Select "None" if applicable. The rational zeros of p(x) :

Solution Steps

To find the zeros of the polynomial \( p(x) = 3x^4 - 14x^3 + 15x^2 + 12x - 20 \), we can use the Rational Root Theorem to identify possible rational zeros. The Rational Root Theorem states that any rational solution, expressed as a fraction \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. We then test these possible zeros by substituting them into the polynomial and checking if they yield zero.

We start with the polynomial: \[ p(x) = 3x^4 - 14x^3 + 15x^2 + 12x - 20 \]

The Rational Root Theorem suggests that any rational root, expressed as \( \frac{p}{q} \), must have \( p \) as a factor of the constant term (-20) and \( q \) as a factor of the leading coefficient (3).

The factors of -20 are \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \). The factors of 3 are \( \pm 1, \pm 3 \).

Possible rational zeros are: \[ \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{5}{3}, \pm \frac{10}{3}, \pm \frac{20}{3} \]

By substituting these possible zeros into the polynomial, we find the actual rational zeros.

The rational zeros of \( p(x) \) are: \[ -1, \frac{5}{3}, 2 \]

Final Answer