Math 41 · Interactive Quiz Adventure · Level Up Your Algebra! 🚀
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🎓 What is a ROOT of a polynomial?
A root (also called a zero) of a polynomial is a value of x that makes the polynomial equal to zero.
If f(c) = 0, then x = c is a root of f(x)
🔍 How to Find Roots
Write the polynomial in factored form. Each factor gives you a root. Set each factor equal to zero and solve!
f(x) = (x − 3)(x + 2)(x − 5)
Roots: x = 3, x = −2, x = 5
✨ What is MULTIPLICITY?
Multiplicity is the number of times a particular root appears as a factor in the polynomial.
f(x) = (x − 2)3(x + 1)2(x − 4)
Root x = 2: multiplicity 3
Root x = −1: multiplicity 2
Root x = 4: multiplicity 1
💡 Quick Rule
The exponent on the factor (x − c)n tells you the multiplicity is n!
📈 Multiplicity & X-Intercepts
The multiplicity tells you how the graph behaves at each x-intercept:
📉📈
ODD Multiplicity
Graph crosses through the x-axis IS an x-intercept ✅
🏀
EVEN Multiplicity
Graph touches and bounces off the x-axis IS still an x-intercept ✅
🌟 Key Insight
All real roots ARE x-intercepts! Complex roots (like imaginary numbers) are NOT x-intercepts. But for real roots: odd multiplicity = crosses, even multiplicity = bounces. Both count as x-intercepts!
🧩 Complex Roots (NOT x-intercepts!)
Some polynomials have factors like (x2 + a2) which give imaginary roots. These have no real x-intercepts!
x2 + 25 = 0 → x = ±5i (imaginary!) NOT an x-intercept ❌
📌 Example 1: From your quiz!
Find the roots and multiplicities of:
g(x) = 130(x + 2)3(x − 6)2(x + 1)2
1
Set each factor = 0: (x+2)=0 → x = −2 | (x−6)=0 → x = 6 | (x+1)=0 → x = −1
2
Read the exponents for multiplicity: (x+2)3 → mult 3; (x−6)2 → mult 2; (x+1)2 → mult 2
3
Odd/Even check: mult 3 = ODD → crosses ✅ x-int; mult 2 = EVEN → bounces ✅ x-int; mult 2 = EVEN → bounces ✅ x-int
✅ Answer:
Root
Multiplicity
X-intercept?
x = −2
3
YES (crosses)
x = 6
2
YES (bounces)
x = −1
2
YES (bounces)
📌 Example 2: From your quiz!
Find the roots and multiplicities of:
f(x) = (x2 + 25)(x2 − 25)
1
Factor x² − 25 = (x−5)(x+5) using difference of squares!
2
Check x² + 25 = 0 → x² = −25 → x = ±5i (imaginary! No real root here)
3
Real roots: x = 5 (mult 1, odd → crosses) and x = −5 (mult 1, odd → crosses)
✅ Answer:
Root
Multiplicity
X-intercept?
x = 5
1
YES (crosses)
x = −5
1
YES (crosses)
x = ±5i
1
NO ❌ (imaginary)
🎯 Choose Your Problems!
Pick any problems you want to practice. You can do them in any order!