Count the crossings

The idea

The graph of a polynomial of degree n intersects the x-axis at at most n points, so it has at most n zeroes; a quadratic has at most 2 and a cubic at most 3.

Degree and crossings

A line has degree 1, a quadratic degree 2, a cubic degree 3. Each zero is a point where the graph meets the x-axis.

Can the graph of a cubic meet the x-axis at 4 points?

• Yes
• No
• Not sure

Count the crossings

Tap every curve. Count the points where it meets the x-axis.

At most n

The graph of a polynomial of degree n meets the x-axis at at most n points. So a polynomial of degree n has at most n zeroes.

A quadratic has at most 2 zeroes and a cubic at most 3. Some cubics have fewer: x³ has only the zero 0, x³ − x² has two (0 and 1), and x³ − 4x has all three (−2, 0 and 2).

Notes

A polynomial of degree n has at most n zeroes: its graph meets the x-axis at at most n points. A quadratic: at most 2. A cubic: at most 3.

Check yourself

What is the largest number of zeroes a cubic polynomial can have?

A polynomial has degree 5. At most how many zeroes can it have?

Answer: 5

A polynomial of degree 5 has at most 5 zeroes.

Ravi says: “x³ meets the x-axis at only one point, so every cubic has just one zero.” What is wrong?

A polynomial has degree 4. Which number of zeroes is impossible?

• 2. x³ − 4x has three zeroes, so a cubic can have more than 2.
• 3 — correct. Yes! Degree 3 means at most 3 zeroes.
• 4. The graph of a cubic meets the x-axis at no more than 3 points.
• any number. The number of zeroes can never be more than the degree.
• A cubic can have up to 3 zeroes, like x³ − 4x with −2, 0 and 2 — correct. Yes! x³ has one zero, but other cubics have more.
• x³ has no zero. x³ has the zero 0, since 0³ = 0.
• Every cubic has exactly 2 zeroes. Some have 1 (x³), some 2 (x³ − x²), some 3 (x³ − 4x).
• 5 — correct. Yes! A polynomial of degree 4 has at most 4 zeroes.
• 4. Possible: at most 4 allows exactly 4.
• 2. Possible: any number up to 4 is allowed.
• 1. Possible: any number up to 4 is allowed.