π Key Concepts You Need to Know
π― What is a Polynomial?
A polynomial is an expression with variables and coefficients using addition, subtraction, and multiplication β no division by variables!
f(x) = aβxβΏ + aβββxβΏβ»ΒΉ + ... + aβx + aβ
The degree is the highest power. The leading coefficient is the number in front of the highest power term.
π End Behavior Rules
End behavior describes what happens as x β +β and x β -β.
- Even degree, positive leading coeff: Both ends go UP ββ
- Even degree, negative leading coeff: Both ends go DOWN ββ
- Odd degree, positive leading coeff: Left DOWN β, Right UP β
- Odd degree, negative leading coeff: Left UP β, Right DOWN β
π± Multiplicity of Roots
The multiplicity of a root is the exponent on its factor.
- Multiplicity 1 (odd): Graph crosses the x-axis βοΈ
- Multiplicity 2 (even): Graph touches (bounces) β¬οΈ
- Multiplicity 3 (odd): Graph crosses with an S-curve γ
f(x) = (xβ2)Β²(x+1)Β³
x=2 has mult. 2 (touches), x=β1 has mult. 3 (crosses with S)
π X & Y Intercepts
X-intercepts (roots/zeros): Set f(x) = 0, solve for x. These are where the graph meets the x-axis.
Y-intercept: Plug in x = 0. This gives f(0).
For f(x) = (x+2)(xβ3): x-int: x=β2, x=3 | y-int: f(0)=(2)(β3)=β6
The total degree = sum of all multiplicities.
βοΈ Worked Example (Just Like Your Homework!)
Analyze and graph:
g(x) = (1/30)(x + 2)Β³(x β 6)Β²(x + 1)Β²
1
Find the degree: 3 + 2 + 2 = 7 (odd degree). Leading coefficient = 1/30 > 0. So: Left end β ββ, Right end β +β
2
Find x-intercepts: Set g(x) = 0 β x = β2 (mult 3, crosses with S-curve), x = 6 (mult 2, bounces), x = β1 (mult 2, bounces)
3
Find y-intercept: g(0) = (1/30)(2)Β³(β6)Β²(1)Β² = (1/30)(8)(36)(1) = 288/30 = 9.6
4
Sketch: Start bottom-left (xβββ, yβββ), cross at x=β2 (with S-shape), bounce at x=β1, cross y-axis at 9.6, bounce at x=6, continue to +β
π― Pick the Problems You Want to Solve!
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