Final answer:
The equation (x^2+9)(x^2+6)= 0 has no complex roots, 0 to 4 real roots, and the imaginary roots are x=±3i and x=±√(6)i.
Step-by-step explanation:
To find the number of complex roots and possible number of real and imaginary roots for the given equation, we will factorize the equation first.
(x^2+9)(x^2+6)= 0
To find complex roots, we need to find the number of times the term "(x^2+9)" and "(x^2+6)" appears. In this case, both terms appear only once, so there are no complex roots.
For real roots, we need to analyze each factor separately. Since both factors are quadratic expressions, they can have at most 2 real roots each. Therefore, the possible number of real roots is 0 to 4.
To find all the roots, we set each factor equal to zero and solve for x:
x^2+9=0 → x^2=-9 → x=±√(-9) → x=±3i
x^2+6=0 → x^2=-6 → x=±√(-6) → x=±√(6)i
So the roots of the equation (x^2+9)(x^2+6)= 0 are x=±3i and x=±√(6)i. These are the imaginary roots.