Cogito
Algebra 2 · Chapter 3 · Lesson 1
Polynomial Behaviour and the Factor Theorem
What the degree tells you before you draw anything.
10 problems · about 21 minutes · A-APR.B.2, A-APR.B.3, F-IF.C.7
What this lesson teaches
The student relates degree and leading coefficient to end behaviour and uses the Factor Theorem.
- 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.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — The Quadratic Formula and the Discriminant: x² + 3x + 2 = 0. What is the discriminant?
Answer 1
Why 1, so two rational roots.
Review — Arithmetic with Complex Numbers: (4 + 3i)(4 − 3i). Result?
Answer 25
Why 25.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x) = x² − 9. What is f(3)?
Answer 0
Why 0, so (x − 3) is a factor.
What does the Factor Theorem say?
Answer f(a) = 0 exactly when (x − a) is a factor.
Why Roots correspond to factors.
Maximum roots of a degree 4 polynomial?
Answer 4
Why The degree.
f(x) = x² − 5x + 6. What is f(2)?
Answer 0
Why 4 − 10 + 6.
Build It Up
The same ideas with more to keep track of.
2 problemsf(x) = x³ − 8. What is f(2)?
Answer 0
Why 8 − 8.
Do the ends of an odd-degree polynomial go the same way?
Answer No. Opposite ways.
Why A cubic goes down then up.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Degree: How many roots can a degree 5 polynomial have at most?
Answer 5
Why 5.
The Test: f(x) = x³ − 7x + 6. What is f(1)?
Answer 0
Why 0, so (x − 1) is a factor.