Use Cauchy's Bound to find an interval \( [a, b] \) which contains all the real zeros of the polynomial \( P(x)=2 x^{4}-10 x^{3}+3 x^{2}-3 x+4 \) : If this is a quiz or exam question, all steps and logic must be shown for full credit. \( \square \) (Use interval notation.)

To find an interval \([a, b]\) that contains all the real zeros of the polynomial \( P(x) = 2x^4 - 10x^3 + 3x^2 - 3x + 4 \) using Cauchy's Bound, we first apply Cauchy's bounds to determine possible brackets for the roots. 1. **Identify the coefficients**: For the polynomial \( P(x) = 2x^4 - 10x^3 + 3x^2 - 3x + 4 \), the coefficients are \( a_0 = 2, a_1 = -10, a_2 = 3, a_3 = -3, a_4 = 4 \). 2. **Using Cauchy's bound**: The bounds can be calculated using the following formula: \[ R = 1 + \max \left( \frac{|a_0|}{|a_n|}, \frac{|a_1|}{|a_n|}, \frac{|a_2|}{|a_n|}, \frac{|a_3|}{|a_n|}, \frac{|a_4|}{|a_n|} \right) \] where \( n \) is the degree of the polynomial, \( a_n \) is the leading coefficient, and \( a_0, a_1, \ldots, a_n \) are the coefficients of the polynomial. 3. **Calculate the bound**: \[ R = 1 + \max \left( \frac{|2|}{2}, \frac{|-10|}{2}, \frac{|3|}{2}, \frac{|-3|}{2}, \frac{|4|}{2} \right) \] The values are: - For \( a_0 = 2 \): \( \frac{|2|}{|2|} = 1 \) - For \( a_1 = -10 \): \( \frac{|-10|}{|2|} = 5 \) - For \( a_2 = 3 \): \( \frac{|3|}{|2|} = 1.5 \) - For \( a_3 = -3 \): \( \frac{|-3|}{|2|} = 1.5 \) - For \( a_4 = 4 \): \( \frac{|4|}{|2|} = 2 \) Taking the maximum of these values, we have: \[ \max(1, 5, 1.5, 1.5, 2) = 5 \] Thus, we calculate: \[ R = 1 + 5 = 6 \] 4. **Specify the intervals**: Cauchy's bound concludes that all real roots of the polynomial \( P(x) \) lie within the interval \([-R, R]\), so: \[ [-6, 6] \] 5. **Final answer**: Therefore, the interval that contains all the real zeros of the polynomial \( P(x) \) is: \[ \boxed{[-6, 6]} \]