Find the zeros of the quadratic function by factoring. What are the \( x \)-intercepts of the graph of the function? : \( P(x)=3 x^{2}-75 \) Select the correct choice below and fill in the answer box to complete your choice. (Use comma to separate answers as needed. Type an integer or a simplfied fraction.) A. The zeros and the \( x \)-intercepts are different. The zeros are B. The zeros and the \( x \)-intercepts are the same. They are

To find the zeros of the quadratic function \( P(x) = 3x^{2} - 75 \), we can start by factoring. First, we factor out the greatest common factor: \[ P(x) = 3(x^{2} - 25) \] Next, we recognize that \( x^{2} - 25 \) is a difference of squares, which we can factor further: \[ P(x) = 3(x - 5)(x + 5) \] Setting \( P(x) \) to zero gives us: \[ 3(x - 5)(x + 5) = 0 \] So, the zeros are: \[ x - 5 = 0 \quad \text{or} \quad x + 5 = 0 \] This gives us: \[ x = 5 \quad \text{and} \quad x = -5 \] Thus, the zeros and the \( x \)-intercepts are the same. They are \( 5, -5 \). B. The zeros and the \( x \)-intercepts are the same. They are 5, -5. --- Did you know that the concept of finding zeros isn't just for polynomials? It's key in economics, physics, and even biology! For instance, economists often set demand or supply equations equal to zero to find equilibrium prices. Additionally, many students stumble at the factoring step by overlooking common factors or jumping right into using the quadratic formula. Remember, if you can factor it neatly, it can save you from the hassle of solving quadratic equations in other forms!