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 $$f(x) = 3x^4 + 2x^2 - 5$$ , the degree is 4.

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

A zero of a polynomial is a value of x that makes the polynomial equal to zero. For example, if $$f(x) = x^2 - 4$$ , then $$x = 2$$ and $$x = -2$$ are zeros.

Multiplicity refers to the number of times a particular zero appears in a polynomial. For example, in $$f(x) = (x - 1)^2(x + 2)$$ , the zero $$x = 1$$ has a multiplicity of 2.

The end behavior of a polynomial function is determined by the degree and the leading coefficient. If the degree is even and the leading coefficient is positive, both ends go up. If the degree is odd and the leading coefficient is positive, the left end goes down and the right end goes up.

The range of a quadratic polynomial of the form $$f(x) = ax^2 + bx + c$$ depends on the value of a. 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