What Is a Polynomial?

A polynomial is a function made by combining powers of $x$ with real-number coefficients. It takes the form:

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

Each term involves a whole-number exponent of $x$, scaled by a real-number coefficient. The exponents must be non-negative integers, and the coefficients can be any real numbers.

The degree of the polynomial is the highest exponent, $n$. This tells you how many turns the graph might have, and how it behaves far out along the number line. The leading coefficient, $a_n$, is the constant attached to the highest power and determines the overall direction of the graph. The constant term, $a_0$, is simply the value of the function when $x = 0$.

Polynomials are smooth and continuous. They don’t have any jumps, breaks, or asymptotes.

End Behaviour

As $x$ becomes very large or very small, the highest-degree term starts to dominate the polynomial. Everything else becomes negligible. In other words,

$$P(x) \sim a_nx^n \quad \text{as } x \to \pm \infty$$

To visualize this, imagine placing a grain of sand on a scale. If that’s the only weight, it matters. But if you place a 100-kilogram object on the scale and then add the grain of sand, the weight barely changes. The sand is still there, but its influence disappears in comparison. That’s how lower-degree terms behave next to the highest-degree term when $x$ gets very large in either direction.

The graph’s end behaviour is determined entirely by two things: the degree $n$ and the leading coefficient $a_n$.

If the degree is even and the leading coefficient is positive, both ends of the graph rise. If the degree is even and the coefficient is negative, both ends fall. If the degree is odd and the coefficient is positive, the left end falls and the right end rises. If the degree is odd and the coefficient is negative, the left end rises and the right end falls.

You don’t need to graph the full function to figure this out—just focus on the term that dominates as $x$ approaches infinity or negative infinity.

Roots and Multiplicity

A root, or zero, of a polynomial is any value of $x$ that makes the function equal zero. That is, $P(r) = 0$. On a graph, these are the $x$-intercepts: places where the curve touches or crosses the horizontal axis.

If $P(r) = 0$, then $(x – r)$ is a factor of the polynomial. This relationship is described by the Factor Theorem, which states that $P(r) = 0$ if and only if $(x – r)$ is a factor of $P(x)$. Solving a polynomial equation, then, is just a matter of finding the graph’s $x$-intercepts.

Sometimes, a root appears more than once. For instance, in the polynomial $P(x) = (x + 2)^2(x – 1)$, the root $x = -2$ has multiplicity 2, while $x = 1$ has multiplicity 1.

In general, if $(x – r)^m$ is a factor of $P(x)$, then $r$ is a root with multiplicity $m$. This multiplicity affects the graph’s shape near the root.

If $m$ is odd, the graph crosses the axis at that point. If $m$ is even, the graph touches the axis and turns back—the function reaches zero but does not change sign. This happens because near $x = r$, the polynomial behaves like $(x – r)^m \cdot Q(x)$, where $Q(x)$ is another polynomial that doesn’t equal zero at $x = r$. If $m$ is odd, then $(x – r)^m$ changes sign as you pass through $r$, causing the overall function to cross the axis. If $m$ is even, it remains the same sign on both sides, so the graph just touches and reverses direction.

You’ll notice this visually. A root of multiplicity 1 creates a sharp crossing of the axis. A multiplicity of 2 results in a bounce, and a multiplicity of 3 still crosses but flattens slightly at the axis.

Dividing Polynomials and the Remainder Theorem

Polynomials can be divided like numbers, either using long division or a faster shortcut known as synthetic division.

Long division works for any two polynomials, but it can be time-consuming. Synthetic division is much quicker when dividing by something simple like $x – c$.

To use synthetic division, you write down just the coefficients of the polynomial and apply a sequence of multiply-and-add steps. Suppose you’re dividing by $x – 3$. You bring down the first coefficient, multiply it by 3, write that result under the next number, and add. You keep repeating that process. The last number in the row is the remainder. The others are the coefficients of the quotient.

If all you need is the remainder, there’s an even faster way. The Remainder Theorem says that if you divide $P(x)$ by $(x – c)$, then the remainder is simply $P(c)$. In other words,

$$P(x) = (x – c)Q(x) + R$$

So when you plug in $x = c$, you get $P(c) = R$. If $P(c) = 0$, then there’s no remainder, and $(x – c)$ is a factor. This ties directly back to the Factor Theorem.

Graphing Polynomial Functions

To sketch a polynomial graph, you don’t need to calculate every point. The key is understanding the structure of the function.

The end behaviour comes from the leading term. The real roots show you where the graph hits the $x$-axis. The multiplicities of those roots tell you whether the graph crosses or bounces. And the turning points show where the graph changes direction. The number of turning points is at most one less than the degree. For a degree-$n$ polynomial, that means a maximum of $n – 1$ turning points. These are found by taking the derivative and locating its roots.

If you plot a few extra values between the roots, you’ll get a better idea of how the curve bends. But even without specific points, you can draw a good sketch just by knowing the intercepts, the end behaviour, and how the graph acts at each root.

The graph is always smooth and doesn’t jump. The sign of the polynomial only changes at roots with odd multiplicity. At even-multiplicity roots, the graph touches the axis but turns around. Complex roots don’t appear directly on the graph, but they still shape the way the graph bends. Since they come in conjugate pairs, their influence shows up in the smooth arcs between visible roots.

A Worked Example

Let’s look at this polynomial:

$$P(x) = 2x^4 – 3x^3 – 11x^2 + 12x + 36$$

This is a degree-4 polynomial, and the leading coefficient is positive. That tells us the ends of the graph both rise.

To find the roots, you might use the Rational Root Theorem. It suggests possible rational roots like ±1, ±2, ±3, and ±3/2. Try some of these using substitution or synthetic division. Suppose you discover that $x = \frac{3}{2}$ is a root. You can factor it out using synthetic division, and then repeat the process with the resulting quotient polynomial.

Eventually, you might reach the fully factored form:

$$P(x) = 2(x + 2)^2(x – \tfrac{3}{2})(x – 3)$$

This tells you everything you need about the graph’s shape. The root at $x = -2$ has multiplicity 2, so the graph touches and turns. The roots at $x = \frac{3}{2}$ and $x = 3$ each have multiplicity 1, so the graph crosses the axis at both of those points.

With three distinct roots and one repeated, you know the graph will bounce at $x = -2$, cross at $x = \frac{3}{2}$, and cross again at $x = 3$. Since it’s degree 4, it can have up to three turning points. You can shape the sketch accordingly. Plot a few extra values if needed to help guide the curve, or use the derivative for more accuracy. But even without those, just knowing the roots and how the graph behaves near them is enough to create a clear, accurate picture.