15. Given that \( \mathrm{x}=1 \) is a zero of the polynomial: \( f(x)=x^{3}+8 x^{2}+11 x-20 \), find the other zeros.

To find the other zeros of the polynomial \( f(x) = x^3 + 8x^2 + 11x - 20 \) given that \( x = 1 \) is a zero, we can perform synthetic division: 1. Set up synthetic division using \( x = 1 \): ``` 1 | 1 8 11 -20 | 1 9 20 ------------------- 1 9 20 0 ``` The remainder is 0, indicating that \( f(x) \) can be factored as \( f(x) = (x - 1)(x^2 + 9x + 20) \). 2. Next, we need to factor the quadratic \( x^2 + 9x + 20 \). We look for two numbers that multiply to 20 and add up to 9. These numbers are 4 and 5. 3. Thus, we can factor the quadratic as: \[ x^2 + 9x + 20 = (x + 4)(x + 5) \] 4. Therefore, the complete factorization of the polynomial \( f(x) \) is: \[ f(x) = (x - 1)(x + 4)(x + 5) \] 5. The other zeros are found by solving \( x + 4 = 0 \) and \( x + 5 = 0 \): - \( x + 4 = 0 \) gives \( x = -4 \) - \( x + 5 = 0 \) gives \( x = -5 \) Thus, the zeros of the polynomial \( f(x) \) are \( x = 1, -4, -5 \).