Penny says that the equation x³-8=0 has only one solution x=2. Use the fundamental theorem of algebra to explain to her why she is incorrect.

The Fundamental Theorem of Algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has precisely n roots, counted with multiplicity, in the complex number system. Penny's equation is x³ - 8 = 0, which is a polynomial of degree 3. According to the theorem, this polynomial should have three roots, not just one.

To find the solutions, the equation can be factored: x³ - 2³ = 0 (x - 2)(x² + 2x + 4) = 0

We can see that one of the roots is x = 2, as Penny suggested. However, we must also consider the second factor x² + 2x + 4. This is a quadratic equation and, as the Fundamental Theorem of Algebra indicates, a quadratic equation has two roots. These roots could be real or complex numbers.

If we proceed to find the roots of the quadratic equation, we might not find real solutions (since the discriminant b² - 4ac = 2² - 4(1)(4) = 4 - 16 = -12 is negative), but we will find two complex solutions. This is done by using the quadratic formula:

x = [-b ± sqrt(b² - 4ac)] / (2a)

Applying this to x² + 2x + 4, one finds the two complex solutions. Therefore, the original polynomial equation has three solutions in total: one real root (x=2) and two complex roots.