Polynomial Functions

Part 14, Chapter 14: Functions and Their Graphs

Learning objectives

Identify the degree and leading coefficient of a polynomial
Evaluate polynomial functions at given points
Describe end behavior based on degree and leading coefficient
Add and multiply polynomials

If you understand polynomials, you understand most of what calculus and algebra ever ask you to compute. They are the simplest functions that can curve, twist, and have multiple roots, and they show up everywhere, fitting data, modelling trajectories, computing taxes, approximating any nice function. Lang treats them early because they are the safest playground: every real number is a legal input, and arithmetic with polynomials never leaves the family.

Definition and vocabulary

A polynomial function is a function of the form

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0,$ cdots+a_1x+a_0,

where the coefficients $a_0, a_1, \ldots, a_n$ are real numbers and $a_n \neq 0$ nneq0. The integer $n$ is the degree, and $a_n$ n is the leading coefficient. The polynomial $f(x) = 7$ has degree $0$ ; $f(x) = 3x + 2$ has degree $1$ (linear); $x^2 - x - 6$ has degree $2$ (quadratic); $x^3$ has degree $3$ (cubic). The domain is all of $\mathbb{R}$ , no division, no roots, nothing breaks.

Closure under add and multiply

If $f$ and $g$ are polynomials, so are $f + g$ and $f \cdot g$ . When you multiply, the degree of the product equals the sum of degrees: $\deg(fg) = \deg(f) + \deg(g)$ . So multiplying a degree-3 by a degree-2 polynomial always lands you in degree 5. This algebraic stability is why polynomials are the natural building blocks of more complicated functions.

Roots and the Fundamental Theorem

A root of $f$ is a value $x_0$ with $f(x_0) = 0$ . Over the reals, a degree- $n$ polynomial has at most $n$ real roots. Over the complex numbers, which you will meet in Chapter 15: the count is exact: every degree- $n$ polynomial has exactly $n$ roots, counted with multiplicity. That statement is the Fundamental Theorem of Algebra, and it is one of the reasons mathematicians wanted to invent complex numbers in the first place.

End behaviour

Far from the origin, a polynomial behaves like its leading term $a_n x^n$ nxn. The sign of $a_n$ n and the parity of $n$ determine the picture: if $n$ is even and $a_n > 0$ n>0 the curve rises on both sides; if $n$ is even and $a_n < 0$ n<0 it falls on both sides; if $n$ is odd the two sides go in opposite directions, with the sign of $a_n$ n telling you which way is up on the right.

Where this shows up

Numerical Analysis: Any continuous function on a closed interval can be approximated to arbitrary accuracy by a polynomial (Weierstrass' theorem); this is why polynomials are the workhorse for curve fitting, ODE solvers, and finite-element methods.

Computer Graphics: Bezier curves, the foundation of vector graphics and font rendering, are polynomial functions of a parameter $t$ ; each glyph in a TrueType font is a low-degree polynomial spline.

Cryptography: Reed-Solomon error-correcting codes (used in CDs, QR codes, deep-space communication) treat data as coefficients of a polynomial; the code is the polynomial evaluated at many points, and any majority can recover the original.

(Set the family to polynomial and drag the four coefficient sliders. Watch how a cubic with a large $a$ and small $b, c, d$ already mimics $a x^3$ at the screen edges, that is the leading-term-wins picture.)

Horner's method for evaluation

To evaluate $2x^3 - x^2 + 3x - 5$ at $x = 4$ , the naive way computes four powers of $4$ separately and adds. Horner's method rewrites the polynomial as $((2x - 1)x + 3)x - 5$ , then evaluates left-to-right: $2 \cdot 4 - 1 = 7$ , then $7 \cdot 4 + 3 = 31$ , then $31 \cdot 4 - 5 = 119$ . Three multiplications and three additions, no extra powers. It is the algorithm every calculator uses internally.

Try it

Predict first: for a cubic to have three real roots, roughly what shape should the graph have (think about local max and local min)? Choose polynomial in the widget, find a setting that yields three real roots, and note the sign pattern on the coefficients.

Predict first: if $a = 0$ , the cubic $ax^3 + bx^2 + cx + d$ collapses to what degree? Set $a = 0$ and confirm the curve crosses the $x$ -axis at most twice (one fewer than a generic cubic).

Evaluate $p(x) = x^3 - 4x^2 + 2x + 5$ at $x = 3$ using Horner's method by hand. Check against the widget's click readout.

Try it in code

A trap to watch for

Students often say a polynomial of degree $n$ has $n$ real roots. It has at most $n$ real roots. The cubic $x^3 + x$ has only one real root (at $x = 0$ ); the other two roots are complex ( $\pm i$ ). The cubic $x^3 - 3x^2 + 3x - 1 = (x - 1)^3$ has only one real root $x = 1$ , but it is a triple root, counted with multiplicity it is three roots, which is consistent with the degree. The right statement is the one above: at most over the reals, exactly over the complexes when counted with multiplicity.

What you now know

You can identify the degree and leading coefficient of any polynomial, predict its end behaviour, add and multiply polynomials without leaving the family, count its real roots correctly using "at most $n$ ," and evaluate it efficiently with Horner's method. The next section visualises a polynomial as the graph of $y = f(x)$ and teaches you to read shape information directly off the picture, intercepts, symmetry, extrema, transformations.

Quick check

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References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 13 §2, the polynomial chapter that anchors this section.

Stewart, J. (2015). Calculus: Early Transcendentals, 8th ed. Cengage. Section 3.2: polynomial functions, end behaviour, and root estimation.

Knuth, D. E. (1997). The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Addison-Wesley. Section 4.6.4, Horner's method and its numerical analysis.

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