Polynomial Functions

Polynomial functions combine powers of x with nonnegative integer exponents and have smooth graphs shaped by degree, zeros, and leading coefficient.

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Zeros connect polynomial factors to x-intercepts. A smooth curve crosses the horizontal axis at three highlighted and labelled zeros, showing how algebraic roots become geometric x-intercepts.

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

P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,

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.

Worked example: build a graph description

These features provide a reliable sketch before any technology is used.

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.

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?