Questions: 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' Rule of Signs, the Quadratic Formula, or other factoring techniques. (Enter your answers as comma-separated lists. Enter all answers including repetitions. If an answer does not exist, enter DNE.) P(x)=4x^4-21x^2+5 rational zeros x= irrational zeros x=

Solution Steps

To find the rational zeros of the polynomial \( P(x) = 4x^4 - 21x^2 + 5 \), we can use the Rational Zeros Theorem, which suggests that any rational zero is a factor of the constant term divided by a factor of the leading coefficient. We will test these possible rational zeros by substituting them into the polynomial. For irrational zeros, we can use techniques such as factoring or the Quadratic Formula on any resulting quadratic expressions.

Using the Rational Zeros Theorem, we find the rational zeros of the polynomial \( P(x) = 4x^4 - 21x^2 + 5 \). The rational zeros identified are: \[ x = -\frac{1}{2}, \quad x = \frac{1}{2} \]

Next, we determine the irrational zeros of the polynomial. The irrational zeros found are: \[ x = \sqrt{5} \approx 2.2361, \quad x = -\sqrt{5} \approx -2.2361 \]

Final Answer