How many solutions does this have?

Question

x³=-x

A) 1

B) 2

C) 3

D) non of the above

Answer 1

x³ = - x

add x for both sides

x³ + x = -x + x

x³ + x = 0

x ( x² + 1) = 0

x=0 or

x² + 1 = 0

x² = -1 , x = ±√ -1 = ± i

so we have three solutions one is real and two are imaginary.

Answer 2

The Fundamental Theorem of Algebra states that for a polynomial of degree N, the total number of real and complex solutions is equal to the degree of the polynomial.

For the equation x³ = x, the degree of the polynomial is equal to 3, therefore there are 3 total solutions to this polynomial.