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 expression. 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, you can look for common factors, use grouping, or apply special factoring formulas such as the difference of squares or the sum/difference of cubes.

The Rational Root Theorem provides a way to find possible rational roots of a polynomial equation. It states that any rational solution, expressed as $$\frac{p}{q}$$ , has p as a factor of the constant term and q as a factor of the leading coefficient.

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

The solutions are $$x = 0, \text{±}2i$$ .

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

The solutions are $$x = \text{±}i \text{√}7, \text{±}\text{√}6$$ .

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