Write the polynomial \( f(x) \) that meets the given conditions. Answers may vary. Degree 3 polynomial with integer coefficients and zeros of \( -\frac{3}{4}, \frac{2}{5} \), and 4 . \( f(x)= \)

To create a degree 3 polynomial with the given zeros, we can express the polynomial in factored form. Since one of the zeros is a fraction, we can multiply each factor by the least common multiple (LCM) of their denominators to obtain integer coefficients. The zeros are \( -\frac{3}{4} \), \( \frac{2}{5} \), and \( 4 \). We can write: \[ f(x) = k \left( x + \frac{3}{4} \right) \left( x - \frac{2}{5} \right) (x - 4) \] To eliminate the fractions, we use the LCM of the denominators \( 4 \) and \( 5 \), which is \( 20 \). Therefore, we multiply each factor by \( 20 \): \[ f(x) = k \left( 20x + 15 \right) \left( 20x - 8 \right) (x - 4) \] Now choosing \( k = 1 \): \[ f(x) = (20x + 15)(20x - 8)(x - 4) \] Let’s expand this step by step. 1. First, expand \( (20x + 15)(20x - 8) \): \[ f(x) = (20x)(20x) + (20x)(-8) + (15)(20x) + (15)(-8) \] \[ = 400x^2 - 160x + 300x - 120 \] \[ = 400x^2 + 140x - 120 \] 2. Now, multiply by \( (x - 4) \): \[ f(x) = (400x^2 + 140x - 120)(x - 4) \] \[ = 400x^3 - 1600x^2 + 140x^2 - 560x - 120x + 480 \] \[ = 400x^3 - 1460x^2 - 680x + 480 \] Thus, the polynomial \( f(x) \) is: \[ f(x) = 400x^3 - 1460x^2 - 680x + 480 \]