Practice Another Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes's Rule of Signs, the Quadratic Formula, or other factoring techniques. (Enter your answers as comma-separated lists. Enter all answers using the appropriate multiplicities. If an answer does not exist, enter DNE.) \[ P(x)=2 x^{4}+5 x^{3}-5 x^{2}-5 x+3 \] rational zeros \( x= \) \( \square \) Irrational zeros \( x= \) \( \square \) CHECK ANSWER

To find the rational zeros of the polynomial \( P(x) = 2x^4 + 5x^3 - 5x^2 - 5x + 3 \), we can use the Rational Zeros Theorem. This theorem states that any rational solution in the form \( p/q \) must have \( p \) as a factor of the constant term (3) and \( q \) as a factor of the leading coefficient (2). The potential rational zeros are: \( \pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2} \). You can apply synthetic division or direct substitution to test these possible zeros. Testing these values, we find the rational zeros: Rational zeros \( x = -3, \frac{1}{2} \) with multiplicities of 1, and DNE. After finding the rational zeros, we can use polynomial division to factor the polynomial into lower-degree terms and look for the irrational zeros with the Quadratic Formula or by further factoring. After completing these steps, the irrational zeros can be determined as well. Irrational zeros \( x = -\frac{1}{4} + \frac{\sqrt{7}}{4}, -\frac{1}{4} - \frac{\sqrt{7}}{4} \) with multiplicities of 1.