Questions: Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). (Enter your answer as an ordered pair or list of numbers.) P(x)=x^4-9 x^3+9 x^2+33 x+14 x=

Solution Steps

To find the real zeros of the polynomial \( P(x) = x^4 - 9x^3 + 9x^2 + 33x + 14 \), we can use the following approach:

1. Factorization: Attempt to factor the polynomial into simpler polynomials, if possible.
2. Quadratic Formula: If the polynomial can be reduced to a quadratic form, use the quadratic formula to find the roots.
3. Numerical Methods: If factorization is not straightforward, use numerical methods such as the Newton-Raphson method or Python's built-in functions to approximate the roots.

The given polynomial is \( P(x) = x^4 - 9x^3 + 9x^2 + 33x + 14 \).

The roots of the polynomial are calculated as follows:

• \( x_1 \approx -1.7808 \)
• \( x_2 \approx -1.0000 \)
• \( x_3 = \frac{1}{7} \)
• \( x_4 \approx 0.2808 \)

Final Answer