Cogito
AP Precalculus · Chapter 1 · Lesson 3
Polynomials of Degree n
What the leading term decides.
11 problems · about 22 minutes · AP Precalculus 1.6, 1.7
What this lesson teaches
The student predicts end behaviour and the maximum number of turning points from a polynomial degree.
- A degree n polynomial has at most n zeros and at most n − 1 turning points.
- End behaviour is decided entirely by the leading term.
- Even degree sends both ends the same way; odd degree sends them opposite ways.
Warm Up
Straightforward practice. Get the method working first.
4 problemsDegree 8. Maximum turning points?
Answer 7
Why 7.
Degree 6. Maximum turning points?
Answer 5
Why n − 1.
Degree 4. Maximum real zeros?
Answer 4
Why At most n.
Match each polynomial to its end behaviour.
Answer x³ → Down on the left, up on the right; −x³ → Up on the left, down on the right; x⁴ → Up at both ends; −x⁴ → Down at both ends
Why Even degree matches the ends; odd degree opposes them.
Build It Up
The same ideas with more to keep track of.
3 problemsDegree 7 with a negative leading coefficient. Does the right end rise? 1 for yes, 0 for no.
Answer 0
Why Odd degree, negative lead.
Minimum real zeros of a degree 4 polynomial?
Answer 0
Why It can float entirely above the axis.
Select every polynomial whose ends go in opposite directions.
Answer x⁵ − x; −3x³
Why Look only at whether the degree is odd.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsDegree 2. Maximum turning points?
Answer 1
Why A parabola turns once.
Which term decides end behaviour? Give the degree of that term for 5x⁴ + 9x⁷ − 2.
Answer 7
Why The highest power, wherever it is written.
Turning Points: A degree 5 polynomial has at most how many turning points?
Answer 4
Why 4.
The Guaranteed Zero: How many real zeros must a degree 3 polynomial have at minimum?
Answer 1
Why At least one.