📈 Polynomial Functions

Part 1 of 7 — Polynomial Basics

1. Polynomials are expressions of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

Polynomials are expressions of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

2. Degree

highest power of x determines the polynomial's behavior

3. Leading coefficient

the coefficient of the highest-degree term

4. Standard form lists terms from highest to lowest degree

Standard form lists terms from highest to lowest degree

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Key Concepts Summary

• Polynomials are expressions of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
• Degree: highest power of x determines the polynomial's behavior
• Leading coefficient: the coefficient of the highest-degree term
• Standard form lists terms from highest to lowest degree

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End Behavior

Part 2 of 7 — End Behavior

1. Even-degree polynomials

both ends go the same direction

2. Odd-degree polynomials

ends go in opposite directions

3. Positive leading coefficient with even degree

both ends up

4. Negative leading coefficient with odd degree

left end up, right end down

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Key Concepts Summary

• Even-degree polynomials: both ends go the same direction
• Odd-degree polynomials: ends go in opposite directions
• Positive leading coefficient with even degree: both ends up
• Negative leading coefficient with odd degree: left end up, right end down

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Zeros & Multiplicity

Part 3 of 7 — Zeros & Multiplicity

1. Zeros (roots)

values of x where f(x) = 0

2. Multiplicity

the number of times a factor repeats

3. Odd multiplicity

graph crosses the x-axis at that zero

4. Even multiplicity

graph touches and bounces off the x-axis

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Key Concepts Summary

• Zeros (roots): values of x where f(x) = 0
• Multiplicity: the number of times a factor repeats
• Odd multiplicity: graph crosses the x-axis at that zero
• Even multiplicity: graph touches and bounces off the x-axis

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Graphing Polynomials

Part 4 of 7 — Graphing Polynomials

1. Plot zeros and y-intercept first

Plot zeros and y-intercept first

2. Use end behavior to sketch the tails

Use end behavior to sketch the tails

3. Apply multiplicity to determine crossing vs bouncing

Apply multiplicity to determine crossing vs bouncing

4. Connect with a smooth, continuous curve

Connect with a smooth, continuous curve

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Key Concepts Summary

• Plot zeros and y-intercept first
• Use end behavior to sketch the tails
• Apply multiplicity to determine crossing vs bouncing
• Connect with a smooth, continuous curve

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Polynomial Division

Part 5 of 7 — Polynomial Division

1. Long division of polynomials

divide step by step

2. Synthetic division

shortcut when dividing by (x - c)

3. Remainder Theorem

f(c) = remainder when dividing by (x - c)

4. Factor Theorem

(x - c) is a factor if and only if f(c) = 0

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Key Concepts Summary

• Long division of polynomials: divide step by step
• Synthetic division: shortcut when dividing by (x - c)
• Remainder Theorem: f(c) = remainder when dividing by (x - c)
• Factor Theorem: (x - c) is a factor if and only if f(c) = 0

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Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

1. Long division of polynomials

divide step by step

2. Synthetic division

shortcut when dividing by (x - c)

3. Remainder Theorem

f(c) = remainder when dividing by (x - c)

4. Factor Theorem

(x - c) is a factor if and only if f(c) = 0

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Key Concepts Summary

• Long division of polynomials: divide step by step
• Synthetic division: shortcut when dividing by (x - c)
• Remainder Theorem: f(c) = remainder when dividing by (x - c)
• Factor Theorem: (x - c) is a factor if and only if f(c) = 0

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Review & Applications

Part 7 of 7 — Review & Applications

1. Long division of polynomials

divide step by step

2. Synthetic division

shortcut when dividing by (x - c)

3. Remainder Theorem

f(c) = remainder when dividing by (x - c)

4. Factor Theorem

(x - c) is a factor if and only if f(c) = 0

Check Your Understanding 🎯

Key Concepts Summary

• Long division of polynomials: divide step by step
• Synthetic division: shortcut when dividing by (x - c)
• Remainder Theorem: f(c) = remainder when dividing by (x - c)
• Factor Theorem: (x - c) is a factor if and only if f(c) = 0

Concept Check 🎯

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