Math101 · Independent Ontario learning library
Math101learn.math101.caPolynomial
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
- 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
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.
Evaluating
Combining polynomials
Add or subtract like terms—terms with the same variable power:
When multiplying, distribute every term and combine afterward:
Polynomials remain polynomials under addition, subtraction, and multiplication.
Zeros and factors
If $P(r)=0$, then $(x-r)$ is a factor. This is the Factor Theorem.
Degree and end behaviour
The leading term controls the long-run graph:
- even degree: both ends point the same way;
- odd degree: the ends point opposite ways;
- positive leading coefficient: the right end rises;
- negative leading coefficient: the right end falls.
Common mistakes
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the degree of 4x⁵ − 2x² + 7?
- The exponents are 5, 2, and 0.
- The greatest exponent is 5.
- The polynomial has degree 5.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
