Questions: If f(x)=x^3+12x^2+41x+30 and x+5 is a factor of f(x), then find all of the zeros of f(x) algebraically.

Solution Steps

To find all the zeros of the polynomial \( f(x) = x^3 + 12x^2 + 41x + 30 \), we can use the fact that \( x+5 \) is a factor. This means \( x = -5 \) is a root of the polynomial. We can perform polynomial division of \( f(x) \) by \( x+5 \) to find the quotient, which will be a quadratic polynomial. Then, we solve the quadratic equation to find the remaining roots.

We start with the polynomial \( f(x) = x^3 + 12x^2 + 41x + 30 \). We know that \( x + 5 \) is a factor, which implies that \( x = -5 \) is a root of the polynomial.

We divide \( f(x) \) by \( x + 5 \) to find the quotient. The result of this division gives us a quadratic polynomial, which we denote as \( g(x) \).

The quotient obtained from the division is \( g(x) = x^2 + 7x + 6 \). We can factor this quadratic as follows: \[ g(x) = (x + 6)(x + 1) \] Setting \( g(x) = 0 \) gives us the additional roots: \[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \] \[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

Combining the known root \( x = -5 \) with the roots from the quadratic, we find all the zeros of the polynomial: \[ x = -5, \quad x = -6, \quad x = -1 \]

Final Answer