A factor (x − 3) gives a zero at x = 3. The factored form hands you every zero directly.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Multiplicity
The exponent on a factor is its multiplicity. In (x − 3)²(x + 1), the zero at 3 has multiplicity 2.
Odd multiplicity crosses
The sign of the output changes, so the curve passes through the axis.
Even multiplicity touches
The sign does not change, so the curve meets the axis and turns back.
Cross or bounce
A zero of odd multiplicity crosses the x-axis; one of even multiplicity touches and turns back. The exponent on a factor is directly visible in the graph, which makes multiplicity readable rather than abstract.
Higher multiplicity flattens
A triple root flattens against the axis before crossing, more so than a simple root. The higher the multiplicity, the more the curve hugs the axis near that zero.
Multiplicities sum to the degree
Counted with multiplicity and including complex roots, a degree-n polynomial has exactly n zeros. Checking that your multiplicities add to the degree confirms you have found them all.
Building a polynomial from its zeros
Zeros at 2 (double) and −1 give (x − 2)²(x + 1), up to a constant factor. One additional point fixes that constant. This construction is a standard exam task and follows straight from the factor theorem.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Sam Read a Factored Polynomial
Sam describes f(x) = (x − 2)²(x + 1) at each of its zeros.
- Step 1
The factors give zeros at x = 2 and x = −1.
Step 4: Your Turn
Practice makes it stick.
The Multiplicity
Problem 1 of 2
In (x − 5)³(x + 2), what is the multiplicity of the zero at x = 5?
Cross or Touch
Problem 2 of 2
At a zero of multiplicity 4, does the graph cross the axis? 1 for yes, 0 for no.
Cross or Bounce
1 of 8
(x − 1)(x + 4). How many distinct zeros?
2 of 8
(x + 3)². Multiplicity of the zero at −3?
3 of 8
Sort each multiplicity by what the graph does at that zero.
Tap something to move it.
- Empty
- Empty
4 of 8
(x − 2)²(x + 5)³. What is the degree?
5 of 8
A zero at x = 0 with multiplicity 1. Does the graph cross? 1 for yes, 0 for no.
6 of 8
(x − 4)⁶. Does the graph cross at 4? 1 for yes, 0 for no.
7 of 8
Select every zero of f(x) = x(x − 3)(x + 2).
8 of 8
Degree 4 with zeros at 1 and 2, both of equal multiplicity. What is each multiplicity?
Step 5: Quick Check
Show what you know.
Question 1 of 1
(x − 7)⁵. Does the graph cross at 7? 1 for yes, 0 for no.
What You Learned
- Each factor gives a zero, and its exponent is that zero’s multiplicity.
- Odd multiplicity crosses the axis; even multiplicity touches and turns back.
- The multiplicities add to the degree.