In short
A polynomial function is f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, where n is a non-negative integer (the degree) and the coefficients a_i are real numbers with a_n \neq 0. The degree and leading coefficient determine the end behaviour — what the graph does as x \to \pm\infty. Linear functions (degree 1), quadratics (degree 2), and cubics (degree 3) are all special cases.
A car leaves Delhi on the highway. For the first hour its odometer reading is roughly 80t km — a linear function of time. But the driver speeds up and slows down, and over four hours a more accurate model might be s(t) = 2t^3 - 9t^2 + 20t. The odometer reading is still a function of time, still built from powers of t — but now the highest power is t^3 instead of t^1. That single change (degree 1 to degree 3) gives the curve a fundamentally different shape: it can bend, change direction, and have an inflection point that a straight line never could.
Lines, parabolas, and this new S-shaped cubic are all members of one family: polynomial functions. The degree of the polynomial — the highest power of x — is the single number that controls how many turns the graph can take and what happens at the far ends. Understanding this family means understanding the skeleton that most of the functions you meet in school are built on.
What makes a function "polynomial"
A polynomial function is a sum of terms, each of which is a constant times a whole-number power of x. No square roots of x, no x in the denominator, no \sin x or 2^x — only non-negative integer exponents.
Polynomial function of degree $n$
A function f is a polynomial function of degree n if it can be written as
where n is a non-negative integer, each a_i is a real number, and a_n \neq 0.
The number a_n is the leading coefficient, a_n x^n is the leading term, and a_0 is the constant term.
Here are the simplest members of the family, listed by degree.
| Degree | Name | Example | Shape |
|---|---|---|---|
| 0 | Constant | f(x) = 5 | Horizontal line |
| 1 | Linear | f(x) = 3x - 2 | Straight line |
| 2 | Quadratic | f(x) = x^2 - 4x + 3 | Parabola (U-curve) |
| 3 | Cubic | f(x) = x^3 - 3x | S-curve (up to 2 turns) |
| 4 | Quartic | f(x) = x^4 - 4x^2 | W-curve (up to 3 turns) |
Linear functions: degree 1
The simplest non-constant polynomial is f(x) = mx + b. Its graph is a straight line with slope m and y-intercept b. There is no bending, no turning — the function either increases everywhere (m > 0), decreases everywhere (m < 0), or stays flat (m = 0, which makes it degree 0).
The key property: the rate of change is constant. For every unit you move to the right, the function value changes by exactly m. That is what "linear" means — the graph is a line.
Quadratic functions: degree 2
You have already met f(x) = ax^2 + bx + c in detail. The graph is a parabola: one turn, one vertex, arms going off to infinity on both sides. The vertex form a(x - h)^2 + k tells you the vertex, and the sign of a decides whether the parabola opens upward (minimum) or downward (maximum).
Two facts from the quadratic expression article that carry over to the polynomial story:
- A degree-2 polynomial has at most 2 real zeros (roots) — the places where the parabola crosses the x-axis.
- The graph has exactly 1 turning point — the vertex.
These numbers — at most 2 zeros, exactly 1 turn — are special cases of a general pattern.
The turning-point rule
A polynomial of degree n has at most n real zeros and at most n - 1 turning points. The word "at most" is important: the polynomial x^2 + 1 has degree 2 but zero real zeros (its parabola floats above the x-axis). The polynomial x^3 has degree 3 but only one real zero and zero turning points (it passes through the origin and never turns back).
Zeros and factored form
The real zeros of a polynomial — the x-values where f(x) = 0 — are exactly the points where the graph touches or crosses the x-axis. If r is a zero, then (x - r) is a factor of the polynomial.
A degree-3 polynomial with three real zeros at x = -2, x = 1, and x = 4 can be written in factored form:
where a controls the vertical stretch. The factored form makes the zeros visible; the expanded form makes the degree and coefficients visible. Both forms describe the same function.
Multiplicity
When a factor appears more than once — like (x - 2)^2 — the zero x = 2 has multiplicity 2. At a zero of odd multiplicity the graph crosses the axis. At a zero of even multiplicity the graph touches the axis and turns back, like a parabola at its vertex.
Take g(x) = (x + 1)(x - 2)^2. The zero at x = -1 has multiplicity 1 (odd), so the graph crosses. The zero at x = 2 has multiplicity 2 (even), so the graph just touches the axis and bounces back.
End behaviour
What does a polynomial do when x is very large (positive or negative)? This is called the end behaviour, and it depends on exactly two things: the degree and the sign of the leading coefficient.
When |x| is large, the leading term a_n x^n overwhelms every other term. Take f(x) = 2x^3 - 100x^2 + 5. At x = 1000, the leading term is 2(10^9) = 2 \times 10^9, while the next term is -100(10^6) = -10^8 — about 20 times smaller. At x = 10{,}000, the ratio is even more lopsided. Far enough from the origin, the polynomial behaves like its leading term alone.
The four possible end behaviours:
| Degree | Leading coefficient | Left end (x \to -\infty) | Right end (x \to +\infty) |
|---|---|---|---|
| Even | Positive (a_n > 0) | \uparrow (up) | \uparrow (up) |
| Even | Negative (a_n < 0) | \downarrow (down) | \downarrow (down) |
| Odd | Positive (a_n > 0) | \downarrow (down) | \uparrow (up) |
| Odd | Negative (a_n < 0) | \uparrow (up) | \downarrow (down) |
The shorthand: even degree — both ends go the same way. Odd degree — ends go opposite ways. The sign of the leading coefficient decides which way.
Think of it this way. For even n, (-x)^n = x^n, so the leading term gives the same sign on both sides — the ends match. For odd n, (-x)^n = -x^n, so the leading term flips sign — the ends oppose.
Higher-degree polynomials
Beyond cubics, the patterns continue. A degree-4 (quartic) polynomial can have up to 4 real zeros and 3 turning points. A degree-5 (quintic) can have up to 5 zeros and 4 turns. Each degree adds one more possible zero and one more possible turn — giving the curve more room to undulate.
But the shape is always smooth — no sharp corners, no jumps, no breaks. A polynomial function is continuous and differentiable everywhere. You can draw its graph without lifting your pen from the paper.
Interactive: exploring end behaviour
Drag the slider to change the leading coefficient a of the cubic f(x) = ax^3. Watch how the end behaviour changes when a switches from positive to negative, and how the steepness changes with |a|.
Example 1: Sketch the graph of $f(x) = -x^3 + 3x^2 - 2x$
Step 1. Factor the polynomial. Every term has a factor of x:
Why: factored form reveals the zeros directly — you can read them off.
Step 2. Identify the zeros. Setting each factor to zero gives x = 0, x = 1, x = 2. Each zero has multiplicity 1, so the graph crosses the axis at all three points.
Why: odd multiplicity means crossing, even multiplicity means touching. All three multiplicities are 1 (odd), so three crossings.
Step 3. Determine the end behaviour. The degree is 3 (odd) and the leading coefficient is -1 (negative). So the graph rises on the left and falls on the right.
Why: odd degree means opposite ends. Negative leading coefficient means: left end up, right end down.
Step 4. Find the turning points. The derivative is f'(x) = -3x^2 + 6x - 2, which equals zero at x = \frac{6 \pm \sqrt{36 - 24}}{6} = \frac{6 \pm \sqrt{12}}{6} = 1 \pm \frac{1}{\sqrt{3}} \approx 0.42 and \approx 1.58.
Plugging in: f(0.42) \approx -0.42(0.42 - 1)(0.42 - 2) = -0.42 \times (-0.58) \times (-1.58) \approx -0.38. And f(1.58) \approx -1.58(0.58)(-0.42) \approx 0.38.
Why: the turning points tell you the local high and low points of the curve, which you need to draw the S-shape accurately.
Step 5. Assemble the sketch. Starting from the left, the curve comes from +\infty, crosses at x = 0, dips to a local minimum near (0.42, -0.38), rises through x = 1, reaches a local maximum near (1.58, 0.38), then crosses at x = 2 and falls to -\infty.
Result. Three zeros at x = 0, 1, 2; a local minimum near x \approx 0.42 and a local maximum near x \approx 1.58; the curve rises left and falls right.
The factored form gave the zeros, the leading term gave the end behaviour, and the turning points filled in the middle. Between these three pieces of information, the shape of the graph is determined.
Example 2: Determine the degree, end behaviour, and zeros of $g(x) = (x - 1)^2(x + 3)$
Step 1. Find the degree. The factor (x - 1)^2 contributes degree 2, and (x + 3) contributes degree 1. Total degree: 2 + 1 = 3.
Why: multiplying the leading terms gives x^2 \cdot x = x^3, confirming degree 3.
Step 2. Find the leading coefficient. Expanding the leading terms: (x)^2 \cdot (x) = x^3. The leading coefficient is 1 (positive).
Why: you only need the highest power of x from each factor to determine the leading coefficient.
Step 3. Determine the end behaviour. Degree 3 (odd), leading coefficient positive. Left end goes down, right end goes up.
Why: odd degree means opposite ends; positive coefficient means the right end goes up.
Step 4. Identify the zeros and their multiplicities. x = 1 (multiplicity 2 — graph touches and turns), x = -3 (multiplicity 1 — graph crosses).
Why: the exponent on each factor is its multiplicity, and multiplicity determines crossing versus touching.
Step 5. Find one extra point for scale. g(0) = (0-1)^2(0+3) = 1 \cdot 3 = 3. So the curve passes through (0, 3).
Result. Degree 3; left end falls, right end rises. Zero at x = -3 (crosses), zero at x = 1 (touches). The y-intercept is 3.
The graph confirms the analysis: the curve touches the axis at x = 1 and turns back (multiplicity 2), crosses at x = -3 (multiplicity 1), and follows the end behaviour predicted by the leading term x^3.
Common confusions
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"The degree tells you exactly how many turning points there are." The degree gives an upper bound: at most n - 1 turns. A cubic can have 2 turning points, but y = x^3 has zero. The actual count depends on the specific coefficients.
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"A polynomial of degree n always has n real zeros." It has at most n real zeros. It always has exactly n zeros if you count complex zeros and count each zero according to its multiplicity (this is the Fundamental Theorem of Algebra) — but the number of real zeros can be less.
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"End behaviour depends on all the coefficients." It depends only on the degree and the leading coefficient. The lower-order terms affect the middle of the graph but become negligible at the extremes.
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"The graph of a polynomial can have a vertical asymptote." Polynomial functions are defined for every real x and are continuous everywhere. No jumps, no gaps, no vertical asymptotes. (Vertical asymptotes belong to rational functions.)
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"A polynomial that touches the x-axis has a zero of multiplicity 1." If the graph touches and turns back without crossing, the zero has even multiplicity (2, 4, ...). Crossing means odd multiplicity.
If you came here to learn how degree and leading coefficient control the shape of a polynomial, you have it — stop here if you like. What follows connects polynomial functions to their algebraic structure.
The Fundamental Theorem of Algebra
Every polynomial of degree n \ge 1 with real (or complex) coefficients has exactly n zeros in the complex numbers, counted with multiplicity. A degree-5 polynomial has exactly 5 complex zeros (some may be real, some may not). The real zeros are the ones you see as x-axis crossings on the graph; the complex zeros are invisible in the real plane.
For polynomials with real coefficients, complex zeros come in conjugate pairs: if 2 + 3i is a zero, then 2 - 3i is also a zero. This means that a polynomial of odd degree must have at least one real zero — since complex zeros pair up, there is always one left over that must be real. Geometrically: an odd-degree polynomial with positive leading coefficient starts below the axis on the left and ends above it on the right, so it must cross the axis at least once.
Building a polynomial from its zeros
If you know the zeros, you can write the polynomial (up to a constant multiple). A cubic with zeros at -1, 2, and 5 is
To pin down a, you need one more piece of information — the y-intercept, a specific function value, or the leading coefficient. For instance, if f(0) = 10, then a(1)(-2)(-5) = 10a = 10, so a = 1.
Polynomial long division and the factor theorem
If f(r) = 0, then (x - r) divides f(x) evenly. This is the Factor Theorem — a direct consequence of the Remainder Theorem (f(x) = (x - r) \cdot q(x) + f(r), and if f(r) = 0, the remainder vanishes). The quotient q(x) has degree one less than f, and you can repeat the process to find all the factors.
Where this leads next
- Polynomials — Introduction — the algebraic foundations: terms, coefficients, addition, and multiplication of polynomials.
- Quadratic Expression and Function — the degree-2 case in full detail: vertex form, axis of symmetry, and extrema.
- Rational Functions — what happens when you divide one polynomial by another. Vertical asymptotes, horizontal asymptotes, and the new kinds of behaviour that fractions produce.
- Polynomial Equations — solving f(x) = 0 for higher-degree polynomials: synthetic division, the rational root theorem, and beyond.
- Graphs of Basic Functions — the toolkit of standard function shapes, of which polynomial functions are one family.