Zeros & Multiplicity

You’ll be able to

Find the real zeros of a factored polynomial
Use multiplicity to decide whether the graph crosses or touches the x-axis
Relate the degree to the total number of zeros counted with multiplicity

Zeros are the x-intercepts

A zero (or root) of a polynomial is an input that makes the output equal to zero — graphically, an x-intercept. When a polynomial is written in factored form, the zeros fall out immediately: each factor (x − r) contributes a zero at x = r. For f(x) = (x − 3)²(x + 1), setting each factor to zero gives x = 3 and x = −1.

Multiplicity: cross or bounce

The multiplicity of a zero is how many times its factor appears. Multiplicity controls the shape at the intercept. An odd multiplicity (1, 3, …) means the graph crosses the x-axis there. An even multiplicity (2, 4, …) means the graph touches the axis and turns back — it “bounces.” In (x − 3)²(x + 1), the zero at x = 3 has multiplicity 2 (a bounce), while the zero at x = −1 has multiplicity 1 (a clean crossing).

Degree = total zeros with multiplicity

deg f = sum of the multiplicities of all zeros

Over the complex numbers a degree-n polynomial has exactly n zeros counted with multiplicity. Adding the exponents of the factors recovers the degree.

Worked example

For f(x) = (x − 3)²(x + 1), list the zeros, give each multiplicity, and state the degree.

1.Set each factor to zero: x − 3 = 0 gives x = 3; x + 1 = 0 gives x = −1.
2.The factor (x − 3) is squared, so the zero x = 3 has multiplicity 2 → the graph bounces there.
3.The factor (x + 1) appears once, so x = −1 has multiplicity 1 → the graph crosses there.
4.Add the multiplicities: 2 + 1 = 3, so the degree of f is 3.

Answer: Zeros: x = 3 (multiplicity 2, bounce) and x = −1 (multiplicity 1, cross). The multiplicities add to 3, so f is a cubic.

Checkpoint

The polynomial f(x) = (x − 3)²(x + 1) has a zero at x = 3. How does the graph behave there?

Watch out

A repeated factor does not create extra separate x-intercepts. (x − 3)² still meets the axis at only one point, x = 3 — the exponent changes the shape (a bounce), not the number of distinct intercepts.

Checkpoint

What is the degree of f(x) = (x − 2)³(x + 5)?

On the exam

When a graph is given, read multiplicity from the shape: a straight-through crossing is multiplicity 1, a flattened crossing is multiplicity 3+, and a bounce is even. This lets you rebuild the factored form directly from a picture.

Answer the 2 checkpoints as you read.