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Find all solutions of the equation (real and/or imaginary) by factoring.
x3+2x2+5x+10=0
To find all solutions of the equation x^3 + 2x^2 + 5x + 10 = 0 by factoring, we first look for any rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (10) divided by the factors of the leading coefficient (1).
Possible rational roots: ±1, ±2, ±5, ±10
By testing these possible rational roots using synthetic division or substitution, we find that none of them are roots of the equation.
Therefore, we can conclude that the equation x^3 + 2x^2 + 5x + 10 = 0 does not have any real roots.
To find complex roots, we can use the Cubic Formula. Let x = a + bi be a complex root of the equation.
Substitute x = a + bi into the equation:
(a + bi)^3 + 2(a + bi)^2 + 5(a + bi) + 10 = 0
Expand and simplify this expression:
a^3 + 3a^2bi - 3ab^2 - b^3i + 2a^2 + 4abi - 2b^2 + 5a + 5bi + 10 = 0
Group the real and imaginary parts:
(a^3 - 3ab^2 + 2a^2 - 2b^2 + 5a + 10) + (3a^2b - b^3 + 4ab + 5b) i = 0
Setting the real and imaginary parts to zero, we get the following system of equations:
a^3 - 3ab^2 + 2a^2 - 2b^2 + 5a + 10 = 0 (1)
3a^2b - b^3 + 4ab + 5b = 0 (2)
Solving this system of equations will give the complex roots of the original equation x^3 + 2x^2 + 5x + 10 = 0.