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, $$2x^3 - 3x^2 + 4 = 0$$ .

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $$4x^5 + 3x^3 - 2$$ , the degree is 5.

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. This can involve techniques such as grouping, using the distributive property, or applying special formulas like the difference of squares.

A root of a polynomial equation is a value of the variable that makes the equation equal to zero. For example, if $$x = 2$$ is a root of $$x^2 - 4 = 0$$ , then substituting 2 into the equation yields 0.

$$x = \frac{3}{2}, \frac{-6 \pm 6i \sqrt{3}}{8}$$ .

$$x = 0, 4$$ .

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

$$x = 0, \pm i \sqrt{3}, \pm \sqrt{3}$$ .

$$x = \frac{3}{2}, \pm 4$$ .