Find all solutions of the equation (real and/or imaginary) by factoring. x^3+2x^2+5x+10=0
To solve this equation by factoring, we need to find a rational root first. We can try to use the Rational Root Theorem to help us find a possible rational root. The theorem says that if a polynomial has a rational root p/q, then p is a factor of the constant term and q is a factor of the leading coefficient.
In this case, the constant term is 10 and the leading coefficient is 1. So the possible rational roots are ±1, ±2, ±5, and ±10.
By trying these values in the equation, we find that the only rational root is x = -2.
Now, we can use synthetic division to divide the polynomial x^3+2x^2+5x+10 by x + 2 (since x = -2 is a root), which gives us:
(x + 2)(x^2 + 2x + 5) = 0
Now, we factor the quadratic equation x^2 + 2x + 5. Since this quadratic does not factor further, we can use the quadratic formula to find the remaining roots:
x = (-2 ± √(2^2 - 4*1*5)) / 2
x = (-2 ± √(4 - 20)) / 2
x = (-2 ± √(-16)) / 2
x = (-2 ± 4i) / 2
Therefore, the solutions to the equation x^3+2x^2+5x+10=0 are:
x = -2 (real root)
x = -1 + 2i (complex root)
x = -1 - 2i (complex root)