Polynomial Functions: Degree, End Behaviour, Zeros and Graphs

What makes a function a polynomial

A polynomial function is a sum of terms, each a constant multiplied by a whole-number power of $x$:

The highest power $n$ is the degree, the coefficient $a_n$ in front of it is the leading coefficient, and $a_0$ is the constant term, which is also the y-intercept. Degree 1 gives lines, degree 2 gives parabolas, degree 3 gives cubics and degree 4 gives quartics. Expressions with negative or fractional powers, such as $\frac{1}{x}$ or $\sqrt{x}$, or with variables in denominators or exponents, are not polynomials.

Polynomials are defined for every real number and their graphs are smooth, unbroken curves with no sharp corners, gaps or asymptotes. That is part of why they are used so widely to model and approximate other functions.

End behaviour from the leading term

For very large positive or negative $x$, the leading term dominates every other term, so the ends of the graph behave like $a_n x^n$. Only two facts matter: whether the degree is even or odd, and whether the leading coefficient is positive or negative.

• Even degree, positive leading coefficient: both ends rise, like $x^2$.
• Even degree, negative leading coefficient: both ends fall, like $-x^2$.
• Odd degree, positive leading coefficient: falls on the left and rises on the right, like $x^3$.
• Odd degree, negative leading coefficient: rises on the left and falls on the right, like $-x^3$.

For example, $p(x) = -2x^5 + 40x^2 + 7$ has odd degree and a negative leading coefficient, so it rises to the left and falls to the right, however large the other coefficients look. An odd-degree polynomial must therefore cross the x-axis at least once, because its two ends point in opposite directions.

Zeros, factors and multiplicity

A zero, or root, of $p$ is a value $c$ with $p(c) = 0$. Zeros are the x-intercepts of the graph, and each real zero $c$ corresponds to a factor $(x - c)$. Factoring is therefore the main tool for finding zeros, using the zero product property: a product is zero only if one of its factors is zero.

If a factor is repeated, the zero has multiplicity greater than one, and the graph behaves differently there. At a zero of odd multiplicity the graph crosses the axis; at a zero of even multiplicity it touches the axis and turns back. The higher the multiplicity, the flatter the graph is near the zero.

A polynomial of degree $n$ has at most $n$ real zeros and at most $n - 1$ turning points, the peaks and valleys of its graph. A cubic can have one or two turning points or none at all, as $x^3$ shows. Methods for finding factors are covered in detail in the guide to factoring polynomials.

The remainder and factor theorems

Dividing a polynomial $p(x)$ by $(x - c)$ leaves a remainder that is just a number, and the remainder theorem says that number is $p(c)$. The factor theorem follows immediately: $(x - c)$ is a factor exactly when $p(c) = 0$.

By contrast, $p(2) = 8 - 8 - 10 + 6 = -4$, so dividing $p(x)$ by $(x - 2)$ would leave a remainder of $-4$, and $(x - 2)$ is not a factor.

Synthetic division

Synthetic division is a compact way to divide by $(x - c)$ using only the coefficients. For Example 3, write $c = 1$ and the coefficients $1, -2, -5, 6$.

1. Bring down the first coefficient: 1.
2. Multiply by $c$ and add to the next coefficient: $1 \times 1 = 1$, and $-2 + 1 = -1$.
3. Repeat: $-1 \times 1 = -1$, and $-5 + (-1) = -6$.
4. Repeat: $-6 \times 1 = -6$, and $6 + (-6) = 0$.
5. The last number, 0, is the remainder. The others, $1, -1, -6$, are the quotient's coefficients: $x^2 - x - 6$.

Every power of $x$ needs a place, so write 0 for any missing term. Dividing $x^3 - 8$ by $(x - 2)$ uses the coefficients $1, 0, 0, -8$ and gives the quotient $x^2 + 2x + 4$ with remainder 0.

Synthetic division works only for divisors of the form $x - c$. To divide by $2x - 1$, divide by $x - \frac{1}{2}$ and then divide the quotient by 2, or use long division.

How many zeros: the fundamental theorem

The fundamental theorem of algebra guarantees that a polynomial of degree $n \ge 1$ has exactly $n$ complex zeros when each is counted as many times as its multiplicity. Some may be real and some complex. When the coefficients are real, complex zeros come in conjugate pairs, so if $2 + 3i$ is a zero, so is $2 - 3i$.

That explains why a cubic with real coefficients always has at least one real zero: its three zeros cannot all be complex, since complex ones arrive in pairs. A quartic, on the other hand, may have no real zeros, as $x^4 + 1$ shows.

When a zero cannot be found exactly, the intermediate value theorem helps locate it. For $p(x) = x^3 + x - 1$, $p(0) = -1$ and $p(1) = 1$. Because polynomial graphs have no gaps, the graph must cross zero somewhere between 0 and 1; numerically the root is about $0.6823$.

Building a polynomial from its zeros

Working backwards, zeros determine a polynomial up to a constant factor. Suppose a quadratic has zeros 2 and $-1$ and passes through $(0, 4)$. It must be $f(x) = a(x - 2)(x + 1)$. Substituting the point gives $4 = a(-2)(1)$, so $a = -2$ and $f(x) = -2(x - 2)(x + 1) = -2x^2 + 2x + 4$. Expanding and checking that $f(0) = 4$, $f(2) = 0$ and $f(-1) = 0$ confirms the result.

Sketching a polynomial graph

1. Find the degree and leading coefficient to fix the end behaviour.
2. Factor to find the zeros and their multiplicities.
3. Mark the y-intercept, the constant term.
4. Decide crossing or touching at each zero from its multiplicity.
5. Join the points with a smooth curve that obeys the end behaviour and has at most $n - 1$ turning points.

This produces an accurate shape, though not the exact heights of turning points, which need calculus. Comparing your sketch with the graphing calculator is a good way to build confidence.

An application: the largest open box

A sheet of card 10 by 16 centimetres has a square of side $x$ cut from each corner, and the sides are folded up to make an open box. Its volume is $V(x) = x(10 - 2x)(16 - 2x) = 4x^3 - 52x^2 + 160x$. Since every length must be positive, the realistic domain is $0 < x < 5$.

The zeros 0, 5 and 8 tell you where the volume collapses to nothing. In between, the volume rises and falls, and calculus pinpoints the maximum: $V'(x) = 12x^2 - 104x + 160 = 4(3x - 20)(x - 2)$, which is zero at $x = 2$ inside the domain. The largest box has volume $V(2) = 2 \cdot 6 \cdot 12 = 144$ cubic centimetres. The derivative rules guide shows how such derivatives are found.

Common mistakes

• Reading end behaviour from the first term written instead of the term of highest degree.
• Leaving out the zero coefficients of missing powers in synthetic division.
• Assuming every zero is a crossing point, ignoring even multiplicity.
• Claiming more turning points than $n - 1$.
• Confusing the zero $c$ with the factor sign: the zero 3 gives the factor $(x - 3)$.

The polynomial calculator finds zeros, factors and divides polynomials with each step shown, so you can check the examples above.

Frequently asked questions