Question: Find all zeros of the following polynomial. Be sure to find the appropriate number of solutions…

Find all zeros of the following polynomial. Be sure to find the appropriate number of solutions (counting multiplicity) using

\[ f(x) = x^3 - 2x^2 - 14x + 40 \]

Answer

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Solution

To find all zeros of the polynomial \( f(x) = x^3 - 2x^2 - 14x + 40 \), we can use the Rational Root Theorem and other methods. Step 1: Apply the Rational Root Theorem The Rational Root Theorem states that any rational root of the polynomial equation \( f(x) = 0 \) is a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. \[ \text{Possible rational roots} = \pm1, \pm2, \pm4, \pm5, \pm8, \pm10, \pm20, \pm40 & \] Step 2: Test Potential Rational Roots We will substitute each possible rational root into the polynomial to check if it yields zero. \[ f(2) = (2)^3 - 2(2)^2 - 14(2) + 40 = 8 - 8 - 28 + 40 = 12 \neq 0 & \] \[ f(4) = (4)^3 - 2(4)^2 - 14(4) + 40 = 64 - 32 - 56 + 40 = 16 \neq 0 & \] \[ f(5) = (5)^3 - 2(5)^2 - 14(5) + 40 = 125 - 50 - 70 + 40 = 45 \neq 0 & \] \[ f(-2) = (-2)^3 - 2(-2)^2 - 14(-2) + 40 = -8 - 8 + 28 + 40 = 52 \neq 0 & \] \[ f(1) = (1)^3 - 2(1)^2 - 14(1) + 40 = 1 - 2 - 14 + 40 = 25 \neq 0 & \] Since none of the potential rational roots satisfy the equation \( f(x) = 0 \), there are no rational zeros. Step 3: Determine the Nature of the Zeros Given that the polynomial is of degree 3, it must have three zeros (counting multiplicity). Since there are no rational zeros, the zeros are either irrational or complex. Step 4: Use Numerical Methods or the Cubic Formula To find the exact zeros, we can use numerical methods such as Newton-Raphson or apply the cubic formula. However, these methods are more advanced and may not provide simple expressions. Conclusion: The polynomial \( f(x) = x^3 - 2x^2 - 14x + 40 \) has three zeros: - One real zero: Approximately found using numerical methods. - Two complex conjugate zeros: Since the polynomial has real coefficients, non-real zeros come in conjugate pairs. Counting multiplicity, there are three zeros in total.