Solving Polynomial Equations

A polynomial equation is an equation that involves a polynomial expression, which is a mathematical expression consisting of variables raised to whole number powers and coefficients. For example, $$x^2 - 4 = 0$$ is a polynomial equation.

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial $$2x^3 - 3x^2 + 4$$ , the degree is 3.

The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n complex roots, counting multiplicities.

To factor a polynomial equation, you express it as a product of its factors. For example, $$x^2 - 4$$ can be factored as $$(x - 2)(x + 2)$$ .

Real roots are solutions to polynomial equations that are real numbers, while complex roots include imaginary numbers. For example, the equation $$x^2 + 1 = 0$$ has complex roots $$x = i$$ and $$x = -i$$ .

The solutions are $$x = \text{±}2i, \text{±}2$$ .

The solutions are $$x = 0, \frac{7}{2}$$ .

The solutions are $$x = \frac{3}{2}, \frac{-6 \text{±} 6i \text{√}3}{8}$$ .

The solutions are $$x = 0, \text{±}i \text{√}3, \text{±}\text{√}3$$ .

The solutions are $$x = \frac{3}{2}, \text{±}4$$ .