Questions: Factor the polynomial f(x). Then solve the equation f(x)=0. f(x)=x^4-3 x^3-6 x^2+28 x-24 The factorization of f(x) is f(x)=

Solution Steps

The polynomial \( f(x) = x^4 - 3x^3 - 6x^2 + 28x - 24 \) can be factored as follows: \[ f(x) = (x - 2)^3 (x + 3) \]

To find the roots of the equation \( f(x) = 0 \), we set the factored form equal to zero: \[ (x - 2)^3 (x + 3) = 0 \] This gives us two factors to consider:

1. \( (x - 2)^3 = 0 \)
2. \( (x + 3) = 0 \)

From \( (x - 2)^3 = 0 \), we find: \[ x - 2 = 0 \implies x = 2 \] This root has a multiplicity of 3.

From \( (x + 3) = 0 \), we find: \[ x + 3 = 0 \implies x = -3 \]

Final Answer