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, $$2x^2 + 3x - 5 = 0$$ .

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $$4x^3 + 2x^2 - x + 7$$ , the degree is 3.

A polynomial equation of degree n can have at most n solutions (real or complex). For example, a cubic polynomial (degree 3) can have up to 3 solutions.

X-intercepts are the points where the graph of the polynomial crosses the x-axis. They are found by setting the polynomial equal to zero and solving for x.

Set the polynomial equal to zero: $$3(x + 4)(x + 1)(x + 3) = 0$$ . The x-intercepts are $$x = -4, -1, -3$$ .

Factoring by grouping is a method used to factor polynomials by grouping terms with common factors. For example, in $$20x^3 - 25x^2 + 4x - 5$$ , group as $$(20x^3 - 25x^2) + (4x - 5)$$ .

The Rational Root Theorem states that any rational solution (or root) of a polynomial equation, in the form $$\frac{p}{q}$$ , has p as a factor of the constant term and q as a factor of the leading coefficient.

The Fundamental Theorem of Algebra states that every non-constant polynomial equation has at least one complex root. This implies that a polynomial of degree n has exactly n roots, counting multiplicities.

To find the x-intercepts, set the equation to zero and solve. This equation can be rewritten as $$(x^2 + 3)^2 = 0$$ , which has no real solutions, hence no x-intercepts.

A complex root is a solution to a polynomial equation that includes imaginary numbers. For example, $$x = 2 + 3i$$ is a complex root.