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Math101learn.math101.caPolynomial Functions
Polynomial functions combine powers of x with nonnegative integer exponents and have smooth graphs shaped by degree, zeros, and leading coefficient.
A polynomial's leading term predicts its distant behaviour, while its zeros and multiplicities organize what happens near the axes.
Definition and notation
A polynomial function has form
where exponents are nonnegative integers and $a_n\ne0$. The highest exponent $n$ is the degree, $a_n$ is the leading coefficient, and $a_0$ is the constant term.
Expressions with $x$ in a denominator, under a radical, or raised to a non-integer power are not polynomials.
Degree and leading behaviour
For large $|x|$, the leading term $a_nx^n$ dominates. Its degree parity and coefficient sign determine end behaviour:
| Degree | Leading coefficient | End behaviour |
|---|---|---|
| even | positive | both ends up |
| even | negative | both ends down |
| odd | positive | left down, right up |
| odd | negative | left up, right down |
Lower-degree terms shape the middle of the graph but cannot change these ultimate directions.
Zeros and factors
A zero $x=r$ satisfies $P(r)=0$ and creates an $x$-intercept $(r,0)$. By the factor theorem, $x-r$ is then a factor.
Factored form
displays zeros $r_i$ and their multiplicities $m_i$. The sum of multiplicities cannot exceed the degree.
Multiplicity and local shape
At a zero with odd multiplicity, the graph crosses the $x$-axis. At a zero with even multiplicity, it touches the axis and turns back. Larger multiplicities make the graph flatter near the intercept.
Multiplicity predicts sign changes: crossing usually changes sign; touching with even multiplicity does not.
Worked example: build a graph description
These features provide a reliable sketch before any technology is used.
Turning points
A degree-$n$ polynomial can have at most $n-1$ turning points. A cubic may have zero or two turning points; a quartic may have one or three, among other possibilities.
This is a maximum count, not a promise. Calculus later locates turning points exactly by solving $P'(x)=0$.
Intercepts and symmetry
The $y$-intercept is always $(0,P(0))=(0,a_0)$. Polynomial functions have domain all real numbers and are continuous, with no gaps, holes, or vertical asymptotes.
If $P(-x)=P(x)$, the function is even and symmetric about the $y$-axis. If $P(-x)=-P(x)$, it is odd and symmetric under a $180^\circ$ rotation about the origin.
Expanding and factoring forms
Expanded form makes degree, leading coefficient, and $y$-intercept easy to read. Factored form makes zeros and multiplicities clear. Neither form is universally “best”; convert based on the question.
Multiplying factors checks an expansion. Dividing by a known factor or using synthetic division can help move back toward factored form.
Transformations
For $y=aP(k(x-d))+c$, transformations follow the general function rules. Vertical shifts change the range and may change all zeros, while horizontal shifts move the entire zero pattern. A negative outside factor reflects across the $x$-axis.
End behaviour must be reconsidered after reflections but remains controlled by the transformed leading term.
Modelling with polynomials
Polynomial models can approximate revenue, volume, trajectories, and smooth trends over restricted intervals. The degree should be justified by structure or data rather than increased simply to force a closer fit.
Outside the fitted interval, high-degree polynomials can behave unrealistically. State the contextual domain and interpret only meaningful roots and extrema.
Common mistakes
Calling the number of terms the degree. Degree is the highest exponent after simplification.
Reading end behaviour from the constant term. Use the leading term.
Assuming every zero crosses. Even multiplicity touches and turns.
Claiming exactly $n-1$ turning points. That is only the maximum.
Treating a graphing-window artefact as an asymptote. Polynomials are continuous and have no vertical or horizontal asymptotes in the usual nonconstant case.
Quick self-check
- What are the degree and leading coefficient?
- What end behaviour follows?
- What are the zeros and their multiplicities?
- Does the graph cross or touch at each zero?
- What is the $y$-intercept?
- Does the sketch respect the turning-point limit and contextual domain?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the end behaviour of P(x) = −2x³ + 5x − 1?
- The leading term is −2x³.
- An odd degree sends the ends in opposite directions.
- A negative leading coefficient gives left up and right down.
End of lesson
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