The degree of a polynomial is its highest exponent, and far from the origin the leading term decides everything. An even degree sends both ends the same way, an odd degree opposite ways, and a negative leading coefficient flips both. Degree n allows at most n − 1 turning points and exactly n roots, counting repeats and complex ones.
Step 1: Let's Learn
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Why the leading term wins
At x = 100, x³ − 4x is 1,000,000 − 400. The lower terms are noise. So x³ − 4x behaves like x³ at the ends: down on the left, up on the right, because a negative cubed is negative.
Four cases
Even and positive: both up, like x². Even and negative: both down, like −x². Odd and positive: left down, right up, like x³. Odd and negative: left up, right down, like −x³. Two questions, four shapes.
Roots from factors
x³ − 4x = x(x − 2)(x + 2), so the real roots are 0, 2 and −2, three crossings for a degree of 3. A factored polynomial shows its roots directly; an unfactored one hides them.
Touch or cross
g(x) = (x − 1)²(x + 3): the root 1 has multiplicity 2, even, so the graph touches the axis at 1 and turns back. The root −3 has multiplicity 1, odd, so it crosses. Total 2 + 1 = 3, the degree.
Between the roots
For x³ − 4x, f(1) = 1 − 4 = −3 and f(−1) = 3. Between roots 0 and 2 the graph dips below the axis; between −2 and 0 it rises above. Two turning points, the most a cubic allows.
Sketching in order
Ends first, from the leading term. Then the roots, marked touch or cross. Then one test value between each pair of roots. That is enough to draw the shape without a calculator.
Step 2: Try It Yourself
Tap and try it out.
Sketch f(x) = x³ − 4x: its ends, its roots and how it meets the axis at each, and where it sits between them
- 1degree 3, odd, leading coefficient +1: left end down, right end upthe leading term x³
Step 0 of 4
Step 3: In Real Life
A roller coaster’s profile
A coaster track is designed as a polynomial. Its degree fixes how many humps it can have, and the leading coefficient whether the ends rise or fall. The designer knows the limits before drawing.
Step 4: 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, even, so both ends go the same way. The leading coefficient is −2, negative, so both ends go down.
Step 5: 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?
7 of 8
Sort each polynomial by its end behavior.
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- Empty
- Empty
8 of 8
A degree-3 polynomial. How many roots in total?
Step 6: 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 behavior?
What You Learned
- Far from the origin the leading term decides the ends: even degree the same way, odd degree opposite ways.
- A negative leading coefficient flips both ends.
- Degree n gives at most n − 1 turning points and exactly n roots, counting repeats and complex ones.
- Odd multiplicity crosses the axis; even multiplicity touches and turns back.