In this lesson:
By the end of this lesson, you should be able to:
You already know quadratics — degree 2 with parabolas:
Today: degree 3, 4, 5, and beyond — same ideas, more structure
Zeros (or roots) of : x-values where
Key strategy: plot zeros first — they anchor everything else
X-intercepts: , ,
Step 1: GCF → Step 2: Difference of squares → Step 3: Zeros: , ,
Use technology: graph in Desmos, then use the "zero" feature
Exact zeros need factoring; technology gives approximate zeros
What are the zeros of ?
Set each factor equal to zero — try it before the next slide.
X-intercepts at , ,
Zeros tell you where the graph hits the x-axis.
End behavior tells you what happens at the edges:
End behavior of as :
All other terms are irrelevant for extreme x-values
Why? At : . The leading term dominates.
Two questions: Even or odd? Positive or negative leading coefficient?
Odd degree, positive leading coefficient:
Odd degree, negative leading coefficient: both directions flip
Key: odd degree → one tail rises, the other falls
Even degree, positive leading coefficient:
Even degree, negative leading coefficient:
Key: even degree → both tails go the same direction
"No matter how complicated the middle, both tails point downward."
What would change if the leading term were ?
What is the end behavior of ?
Two questions: even or odd degree? Positive or negative leading coefficient?
Find zeros and state end behavior for each:
Work through each, then check the next slide.
1. Zeros at ; degree 3, positive → falls left, rises right
2. Zeros at ; degree 3, negative → rises left, falls right
3. Degree 4, positive → rises on both ends
✓ Zeros come from factored form — set each factor to zero
✓ Only degree and leading coefficient determine end behavior
Watch out: middle terms do NOT affect end behavior
Watch out: a degree- polynomial has AT MOST zeros
Lesson 2 of 2:
Combine everything into a complete polynomial sketch.