Graphs 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 zeros (or roots) of a polynomial are the values of x for which the polynomial evaluates to zero. For example, if $$f(x) = (x - 1)(x + 2)$$ , the zeros are x = 1 and x = -2.

To find the zeros of a polynomial, set the polynomial equal to zero and solve for x. For example, for $$f(x) = x^2 - 5x + 6$$ , set $$x^2 - 5x + 6 = 0$$ and factor or use the quadratic formula.

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

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

The factored form of a polynomial expresses it as a product of its factors. For example, $$f(x) = (x - 2)(x + 3)$$ is the factored form of the polynomial.

A polynomial of degree n can have at most n zeros. For example, a quadratic polynomial (degree 2) can have up to 2 zeros.

The number of zeros of a polynomial is related to its degree. A polynomial of degree n can have up to n real zeros, counting multiplicities.

A polynomial function is a function that can be represented by a polynomial equation. For example, $$f(x) = 2x^2 + 3x - 5$$ is a polynomial function.

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