Math101 · Independent Ontario learning library
Math101Polynomial
A polynomial combines constant multiples of whole-number powers into one of algebra's most useful families.
A polynomial is a finite sum of terms $a_kx^k$ whose exponents are non-negative integers.
Anatomy
Consider
$$
P(x)=3x^4-2x^2+7x-5.
$$
- Terms: $3x^4$, $-2x^2$, $7x$, and $-5$.
- Leading term: $3x^4$.
- Leading coefficient: $3$.
- Constant term: $-5$.
- Degree: $4$, the greatest exponent with a nonzero coefficient.
Missing powers are allowed. Here the coefficient of $x^3$ is zero even though the term is not written.
Formal form
$$
P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,
$$
where $n$ is a non-negative integer and $a_n\ne0$.
What is not a polynomial
- $x^{-2}+1$: negative exponent;
- $\sqrt{x}+3$: fractional exponent;
- $1/(x-1)$: variable in a denominator;
- $2^x$: variable in an exponent.
Irrational coefficients are allowed. For example, $\sqrt2x^3$ is a polynomial.
Names by term count
These labels describe the number of terms, not the degree.
