Characteristics of Polynomials

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, $$f(x) = 2x^3 - 4x^2 + 3x - 5$$ is a polynomial.

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

The domain of a polynomial function is all real numbers, represented as $$(-\text{∞}, \text{∞})$$ .

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. For example, for $$f(x) = x^2 - 4x + 3$$ , the zeros are $$x = 1$$ and $$x = 3$$ .

A relative minimum is a point on the graph of a polynomial where the function value is lower than the values of the function at nearby points. For example, if $$f(x)$$ has a relative minimum at point $$(0, -2)$$ , then $$f(0) = -2$$ is less than the values of $$f(x)$$ around it.

An increasing interval is a range of x-values where the polynomial function is rising. For example, if a polynomial is increasing on the interval $$(-2, 0)$$ and $$(2, \text{∞})$$ , it means that as x moves through these intervals, the function values increase.

A decreasing interval is a range of x-values where the polynomial function is falling. For example, if a polynomial is decreasing on the interval $$(-\text{∞}, -2)$$ , it means that as x moves through this interval, the function values decrease.

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. For example, in $$f(x) = 4x^3 + 2x^2 - 5$$ , the leading coefficient is 4.

The standard form of a polynomial is written in descending order of the degree of its terms. For example, $$f(x) = 3x^4 - 2x^3 + x - 7$$ is in standard form.

A monomial is a polynomial with one term (e.g., $$3x^2$$ ), a binomial has two terms (e.g., $$x^2 - 4$$ ), and a trinomial has three terms (e.g., $$x^2 + 3x + 2$$ ).