A polynomial is an algebraic expression built from terms whose variable exponents are nonnegative integers. For example,

$$P(x)=3x^2-2x+5$$

is a polynomial in $x$.

Degree and coefficients

The numbers multiplying the powers of the variable are the coefficients. In the example above, the coefficients are $3$, $-2$ and $5$.

The largest exponent with a nonzero coefficient is the degree of the polynomial, so $3x^2-2x+5$ has degree $2$. The term with the highest power, here $3x^2$, is the leading term.

Arithmetic with polynomials

Polynomials can be added and subtracted by combining terms with the same power. They can be multiplied using the distributive property. For example,

$$(x+2)(x+3)=x^2+5x+6.$$

The result is again a polynomial.

Evaluating polynomials and zeros

A polynomial can be evaluated by substituting a value for its variable. For

$$P(x)=x^2-4,$$

we have $P(3)=5$.

A value $r$ is a zero of a polynomial when $P(r)=0$. In this example,

$$P(2)=0,\qquad P(-2)=0,$$

so $2$ and $-2$ are zeros of $P$.