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Roots, bounces and ends

Polynomial graphs, visually

A polynomial is decided by three things: its roots, how it meets the ground at each root, and where its two ends go. Drag the roots and watch the track change, then make two of them the same.

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y = (x + 2)(x − 1)(x − 3)
  1. degree 3, odd: the ends point opposite ways
  2. crosses at x = −2, 1, 3
  3. at most 2 turning points
Three different roots, so the track crosses the ground three times.
Idea 1 · The boss term and the two ends

The boss term and the two ends

The degree is the highest power of x, and the leading coefficient is the number in front of it. Write the terms in order first, so you do not take the first one written.

Together they say where the two ends go. Even degree: both ends point the same way. Odd degree: they point opposite ways. A negative number in front flips the ends.

Aha!

The boss term decides where the two arms point: the degree says same or opposite, the sign says up or down.

Idea 2 · The stations

Roots from the factors: the stations

In factored form each bracket gives a root: (x − 3) gives 3 and (x + 2) gives −2. These are the stations where the track meets the ground.

Set each bracket to zero and solve. The sign in the bracket is the sign slip to watch for: (x + 2) gives −2, not 2.

Aha!

Each bracket is a station on the ground: set it to zero to find where the track meets the x-axis.

Idea 3 · Cross or bounce

Cross or bounce: multiplicity

If a bracket appears once, the graph crosses the x-axis at that root. If it appears twice, as in (x − 1)², the graph touches the axis and bounces back.

How many times a root counts is its multiplicity. Odd multiplicities cross, even multiplicities bounce. Not every root crosses.

Aha!

A root that counts twice only touches the ground and turns back, while a single root crosses.

Idea 4 · Above or below

Above or below between the roots

Between two neighboring roots the graph stays on one side of the ground, so its sign cannot change there. Test one point in each stretch to see if it is above or below.

A polynomial of degree n has at most n − 1 turning points, the hills and dips. A cubic has at most 2.

Aha!

Between two roots the track stays on one side of the ground, so one test point tells you the whole stretch.

Quick answers

Questions about polynomials

What is the degree of a polynomial?

The highest power of x. The leading coefficient is the number in front of that term. For 2x⁵ − x + 7 the degree is 5 and the leading coefficient is 2.

How do you find the end behavior of a polynomial?

Look at the leading term. With an even degree both ends point the same way: up if the number in front is positive. With an odd degree the ends point opposite ways, and the right end goes up if the number in front is positive.

What is the multiplicity of a root?

How many times its bracket appears in the factored form. A root with odd multiplicity crosses the x-axis. A root with even multiplicity touches the axis and turns back.

How many turns can a polynomial graph have?

At most one fewer than its degree. A cubic has at most 2 turning points, and a degree-4 polynomial has at most 3. It can have fewer.

Now play it

Try it in Pip: Math Adventure

World 6 of Pip: Math Adventure is about polynomials. You lay a track through the stations, and the roots and the two ends decide what the track does.

It is a math game for iPhone, coming soon to the App Store. Join the list and we will tell you the day Pip is out.

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