Sure, let's find all the real zeros of the polynomial [tex]\( x(x) = x^4 - 9x^3 + 9x^2 + 33x + 14 \)[/tex]. We'll proceed step-by-step for clarity.
### Step 1: Identify the Polynomial
We are given the polynomial:
[tex]\[ P(x) = x^4 - 9x^3 + 9x^2 + 33x + 14 \][/tex]
### Step 2: Find the Zeros of the Polynomial
We need to find the real zeros of this polynomial, which means finding the values of [tex]\( x \)[/tex] for which [tex]\( P(x) = 0 \)[/tex]. The equation thus becomes:
[tex]\[ x^4 - 9x^3 + 9x^2 + 33x + 14 = 0 \][/tex]
### Step 3: Factor the Polynomial (if possible)
Generally, solving for the roots of higher-degree polynomials can be challenging, and they may not always factor neatly into simpler expressions. In such cases, methods such as the Rational Root Theorem, synthetic division, or more advanced algebraic techniques might be used. In some cases, numerical methods or the use of symbolic computation software (which will not be shown here) can be employed.
### Step 4: Solve the Polynomial Equation
For this specific polynomial,
[tex]\[ x^4 - 9x^3 + 9x^2 + 33x + 14 = 0 \][/tex]
We can solve it directly:
[tex]\[
\begin{aligned}
\text{The real zeros of the polynomial are:} & \\
x_1 = -1, & \\
x_2 = 7, & \\
x_3 = \frac{3}{2} - \frac{\sqrt{17}}{2}, & \\
x_4 = \frac{3}{2} + \frac{\sqrt{17}}{2}. & \\
\end{aligned}
\][/tex]
### Step 5: Present the Zeros
So, the real zeros of the polynomial [tex]\( P(x) = x^4 - 9x^3 + 9x^2 + 33x + 14 \)[/tex] are:
[tex]\[ -1, \, 7, \, \frac{3}{2} - \frac{\sqrt{17}}{2}, \, \frac{3}{2} + \frac{\sqrt{17}}{2} \][/tex]
These values of [tex]\( x \)[/tex] satisfy the equation [tex]\( P(x) = 0 \)[/tex] and are the real roots of the polynomial.
