Given the function \( C(x)=4 x^{4}+20 x^{3}+24 x^{2} \) its \( C \)-intercept is its \( x \)-intercepts are

To find the \( C \)-intercept and the \( x \)-intercepts of the function \( C(x) = 4x^4 + 20x^3 + 24x^2 \), we can follow these steps: ### 1. Finding the \( C \)-intercept: The \( C \)-intercept occurs when \( x = 0 \). We can find it by evaluating \( C(0) \): \[ C(0) = 4(0)^4 + 20(0)^3 + 24(0)^2 = 0 \] Thus, the \( C \)-intercept is \( (0, 0) \). ### 2. Finding the \( x \)-intercepts: The \( x \)-intercepts occur when \( C(x) = 0 \). We need to solve the equation: \[ 4x^4 + 20x^3 + 24x^2 = 0 \] We can factor out the common term \( 4x^2 \): \[ 4x^2(x^2 + 5x + 6) = 0 \] Setting each factor to zero gives us: 1. \( 4x^2 = 0 \) which leads to \( x^2 = 0 \) or \( x = 0 \). 2. \( x^2 + 5x + 6 = 0 \) Now, we can solve the quadratic equation \( x^2 + 5x + 6 = 0 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 5 \), and \( c = 6 \): \[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} = \frac{-5 \pm \sqrt{25 - 24}}{2} = \frac{-5 \pm 1}{2} \] Calculating the two possible values: 1. \( x = \frac{-5 + 1}{2} = \frac{-4}{2} = -2 \) 2. \( x = \frac{-5 - 1}{2} = \frac{-6}{2} = -3 \) ### Summary: - The \( C \)-intercept is \( (0, 0) \). - The \( x \)-intercepts are \( x = 0, -2, -3 \). Thus, the \( x \)-intercepts are \( (0, 0) \), \( (-2, 0) \), and \( (-3, 0) \).