Let's solve the cubic equation given by:
[tex]\[ x^3 - 7x^2 + 15x - 25 = 0 \][/tex]
We are provided with the information that the real root of this equation is [tex]\( x = 5 \)[/tex]. Our task is to find the nonreal (complex) roots of the equation.
First, we can confirm that [tex]\( x = 5 \)[/tex] is a root by substituting it into the equation:
[tex]\[ 5^3 - 7 \cdot 5^2 + 15 \cdot 5 - 25 = 125 - 175 + 75 - 25 = 0 \][/tex]
Since [tex]\( x = 5 \)[/tex] is indeed a root, we can factor out [tex]\( (x - 5) \)[/tex] from the original polynomial.
Next, we perform polynomial division of [tex]\( x^3 - 7x^2 + 15x - 25 \)[/tex] by [tex]\( x - 5 \)[/tex] to find the quadratic factor.
Once we obtain the quadratic factor, we will solve the quadratic equation for the other roots. However, based on the real root [tex]\( x = 5 \)[/tex] and assuming the form of the remaining solutions being complex conjugates, we can identify the correct choice directly.
The nonreal roots for the polynomial equation can be found to be:
[tex]\[ 1 + 2i \][/tex]
[tex]\[ 1 - 2i \][/tex]
To summarize, the nonreal roots of the equation [tex]\( x^3 - 7x^2 + 15x - 25 = 0 \)[/tex] are:
[tex]\[ 1 + 2i \][/tex]
[tex]\[ 1 - 2i \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{1+2i, 1-2i} \][/tex]
