The degree is the highest power. It caps the number of roots and sets the end behaviour.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What the ends do
Even degree sends both ends the same way. Odd degree sends them opposite ways.
The Factor Theorem
If f(a) = 0 then (x − a) is a factor, and the other way round. Roots and factors are the same information.
Repeated roots
A root appearing twice makes the graph touch the axis and turn back rather than cross.
The degree bounds the behaviour
A degree-n polynomial has at most n roots and at most n − 1 turning points. Knowing the degree tells you the maximum complexity of the graph before you plot a single point.
End behaviour comes from the leading term
For large |x| the highest-degree term dominates everything else. Even degree with positive leading coefficient rises at both ends; odd degree rises at one end and falls at the other. Two facts fix the shape at the extremes.
The factor theorem
If f(a) = 0 then (x − a) is a factor, and conversely. Roots and factors are the same information in two notations, which is why finding one root immediately reduces the degree of what remains.
Multiplicity shapes the crossing
A root of odd multiplicity crosses the axis; a root of even multiplicity touches and turns back. Higher multiplicity flattens the graph near the root. The exponent on a factor is visible in the picture.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 2)
Step 3: Watch an Example
One step at a time.
Watch Ravi Use the Factor Theorem
f(x) = x³ − 4x² + x + 6. Ravi tests x = 3.
- Step 1
He substitutes: 27 − 36 + 3 + 6.
Step 4: Your Turn
Practice makes it stick.
The Degree
Problem 1 of 2
How many roots can a degree 5 polynomial have at most?
The Test
Problem 2 of 2
f(x) = x³ − 7x + 6. What is f(1)?
Degree, Roots, Factors
1 of 4
Maximum roots of a degree 4 polynomial?
2 of 4
f(x) = x² − 5x + 6. What is f(2)?
3 of 4
f(x) = x³ − 8. What is f(2)?
4 of 4
Do the ends of an odd-degree polynomial go the same way?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x) = x² − 9. What is f(3)?
Question 2 of 2
What does the Factor Theorem say?
What You Learned
- The degree caps the number of roots and sets the end behaviour.
- f(a) = 0 exactly when (x − a) is a factor.
- A repeated root touches the axis rather than crossing it.