The degree of a polynomial is its highest exponent. It controls almost everything about the graph shape.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
End behaviour
Far from the origin the leading term dominates. An even degree sends both ends the same way; an odd degree sends them opposite ways.
The leading coefficient
A negative leading coefficient flips both ends. Degree and sign together fix the end behaviour completely.
Turning points
A polynomial of degree n has at most n − 1 turning points. It may have fewer, never more.
Roots
A polynomial of degree n has exactly n roots, counting repeats and complex ones. Real roots are the x-axis crossings.
Multiplicity
At an odd multiplicity the graph crosses the axis. At an even one it touches and turns back.
The degree bounds the complexity
A degree-n polynomial has at most n real roots and at most n − 1 turning points. Knowing the degree caps what the graph can do before any plotting.
The leading term decides the ends
For large |x| the highest-degree term dominates. Even degree with positive leading coefficient rises at both ends; odd degree goes opposite ways. Two facts fix both extremes.
Multiplicity shapes each crossing
Odd multiplicity crosses the axis; even multiplicity touches and turns back. Higher multiplicity flattens the curve near the root, so the exponent on a factor is visible in the picture.
Polynomials are the well-behaved family
Continuous and smooth everywhere, with no asymptotes, holes or corners. That makes them the baseline against which rational, radical and trigonometric behaviour is compared.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Rosa Describe a Quartic
Rosa analyses f(x) = −2x⁴ + 3x² − 1.
- Step 1
The degree is 4, which is even, so both ends go the same way.
Step 4: Your Turn
Practice makes it stick.
The Turns
Problem 1 of 2
A degree-5 polynomial. At most how many turning points?
The Roots
Problem 2 of 2
A degree-5 polynomial. How many roots, counting repeats and complex ones?
Read the Polynomial
1 of 8
f(x) = 3x⁴. How many ends go up?
2 of 8
f(x) = −3x⁴. How many ends go up?
3 of 8
f(x) = 2x⁵. How many ends go up?
4 of 8
A degree-6 polynomial. At most how many turning points?
5 of 8
(x − 3)²(x + 1) = 0. Multiplicity of the root 3?
6 of 8
At an even multiplicity, does the graph cross the axis? 1 yes, 0 no.
7 of 8
Sort each polynomial by its end behaviour.
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- Empty
- Empty
8 of 8
A degree-3 polynomial. How many roots in total?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A degree-4 polynomial. At most how many turning points?
Question 2 of 2
What decides end behaviour?
What You Learned
- Even degree sends both ends the same way; odd degree sends them opposite ways.
- A negative leading coefficient flips both ends.
- Degree n gives at most n − 1 turning points and exactly n roots.