Question

State the number of complex roots and the possible number of real and imaginary roots for the equation: x^2(x+2)(x-5) = 0.

a) 2 complex roots, 1 real root
b) 4 complex roots
c) 2 complex roots, 2 real roots
d) 1 complex root, 2 real roots

Answer 1

The equation x^2(x+2)(x-5) = 0 has four real roots (x=0 with multiplicity two, x=-2, and x=5), meaning it has four complex roots, as all real roots are also complex. There are no imaginary roots, so the correct answer is (c) 2 complex roots, 2 real roots.

Step-by-step explanation:

The given equation is x^2(x+2)(x-5) = 0. To state the number of complex roots and the possible number of real and imaginary roots, we need to analyze the factors of the equation individually.

The equation can be broken down into its factors: x^2, (x+2), and (x-5). Each factor set to zero gives us a root of the equation:

• x^2 set to zero gives us one real root at x = 0, but since it is squared, it counts as two identical real roots.
• (x + 2) set to zero gives us another real root at x = -2.
• (x - 5) set to zero gives us another real root at x = 5.

Therefore, we have a total of 4 real roots. Since all our roots are real and no coefficient has an imaginary part, there are no imaginary roots. Remember that all real numbers are also complex numbers, so we have 4 complex roots which happen to be real in this case. The correct answer, accounting for the real and complex roots, is (c) 2 complex roots, 2 real roots.