Polynomial Key Features

The degree of a polynomial is the highest power of the variable in the polynomial expression.

If a polynomial has an odd degree, its graph will have opposite end behaviors; as x approaches positive infinity, the polynomial approaches positive infinity, and as x approaches negative infinity, it approaches negative infinity.

If a polynomial has an even degree, its graph will have the same end behaviors; as x approaches both positive and negative infinity, the polynomial approaches either positive or negative infinity.

The leading coefficient is the coefficient of the term with the highest degree in a polynomial.

If the leading coefficient is positive, the graph rises to the right; if it is negative, the graph falls to the right.

Zeros of a polynomial are the values of x for which the polynomial equals zero; they are also the x-intercepts of the graph.

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n zeros, counting multiplicities.

The factored form of a polynomial expresses it as a product of its linear factors.

A double root means that the factor corresponding to that root appears twice in the factored form, which affects the graph by causing it to touch the x-axis and turn around.

The end behavior can be determined by the degree (odd/even) and the sign of the leading coefficient (positive/negative).