Find all solutions of the equation (real and/or imaginary) by factoring.
x^3+2x^2+5x+10=0
To factor the given equation, we start by looking for a factor of the constant term (10) that sums up to the coefficient of the linear term (5).
The possible pairs of factors of 10 are: (1,10), (2,5), (-1,-10), (-2,-5)
From the pairs of factors, we see that the pair that sums up to 5 is (2,5).
Next, we rewrite the equation using this information:
x^3 + 2x^2 + 5x + 10 = 0
x^3 + 2x^2 + 2x + 3x + 10 = 0
x^2(x + 2) + 1(x + 2) = 0
(x^2 + 1)(x + 2) = 0
Set each factor equal to zero and solve for x:
x^2 + 1 = 0
x^2 = -1
x = ±√(-1)
x = ±i
x + 2 = 0
x = -2
Therefore, the solutions to the equation x^3 + 2x^2 + 5x + 10 = 0 are x = -2, x = i, and x = -i.