📐 Concept diagram

03-03 - Polynomial Functions

Phase: 3 | Subject: 03-03 Prerequisites: 01-07-quadratic-expressions.md, 01-08-quadratic-equations.md, 03-02-transformations-of-functions.md Next subject: 03-04-rational-functions.md

Learning Objectives

By the end of this subject, you will be able to:

1. Identify degree, leading coefficient, and end behaviour
2. Sketch polynomial graphs showing turning points
3. Find roots and understand their multiplicities
4. Solve polynomial inequalities
5. Apply the Rational Root Theorem

Core Content

Polynomial Terminology

A polynomial has the form: $P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀$

• Degree: highest power n
• Leading coefficient: aₙ
• Constant term: a₀

Examples: - 3x⁴ - 2x² + 5: degree 4, leading coefficient 3 - -x³ + 4x: degree 3, leading coefficient -1 - 7: degree 0 (constant polynomial)

End Behaviour

Determined by degree (n) and leading coefficient (aₙ):

Memory: Odd degree: ends go opposite directions. Even degree: ends go same direction.

Roots and Multiplicity

A root (or zero) is where P(x) = 0. Graphically, x-intercepts.

Multiplicity = how many times a root is repeated.

• Odd multiplicity (1, 3, 5...): graph CROSSES the x-axis
• Even multiplicity (2, 4, 6...): graph TOUCHES and turns back

Example: P(x) = (x - 1)³(x + 2)² - Root x = 1: multiplicity 3 (odd) — crosses - Root x = -2: multiplicity 2 (even) — touches and turns

Turning Points

A polynomial of degree n has at most n - 1 turning points.

Example: Degree 4 → at most 3 turning points.

Rational Root Theorem

If P(x) = aₙxⁿ + ... + a₀ has integer coefficients and a rational root p/q (in lowest terms): - p divides a₀ (constant term) - q divides aₙ (leading coefficient)

Example: P(x) = 2x³ - 3x² - 11x + 6 Possible rational roots: ±1, ±2, ±3, ±6, ±1/2, ±3/2

Testing: P(1) = 2 - 3 - 11 + 6 = -6 ≠ 0 P(2) = 16 - 12 - 22 + 6 = -12 ≠ 0 P(3) = 54 - 27 - 33 + 6 = 0 ✓

So (x - 3) is a factor.

Key Terms

• Multiplicity

Worked Examples

Example 1: End behaviour and sketch

P(x) = -x⁴ + 2x² - 1

• Degree 4 (even), leading coefficient -1 (negative)
• Both ends DOWN
• Symmetric about y-axis (only even powers)
• y-intercept: -1
• Roots: solve -x⁴ + 2x² - 1 = 0 → x⁴ - 2x² + 1 = 0 → (x² - 1)² = 0 → x = ±1 (multiplicity 2)

Graph touches at x = ±1, both ends go down.

Example 2: Find all roots

P(x) = x³ - 6x² + 11x - 6

Possible roots: ±1, ±2, ±3, ±6 P(1) = 1 - 6 + 11 - 6 = 0 ✓

Divide by (x - 1): x² - 5x + 6 Factor: (x - 2)(x - 3)

Roots: x = 1, 2, 3 (all multiplicity 1)

Quiz

Q1: What does the concept of Multiplicity primarily refer to in this subject?

A) A historical anecdote about Multiplicity B) A computational error related to Multiplicity C) The definition and application of Multiplicity D) A visual representation of Multiplicity

Correct: C)

• If you chose A: This is incorrect. Multiplicity is defined as: the definition and application of multiplicity. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Multiplicity is defined as: the definition and application of multiplicity. The other options describe different aspects that are not the primary focus.
• If you chose C: Multiplicity is defined as: the definition and application of multiplicity. The other options describe different aspects that are not the primary focus. Correct!
• If you chose D: This is incorrect. Multiplicity is defined as: the definition and application of multiplicity. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Polynomial Terminology?

A) It is primarily a historical notation system B) It is used only in advanced research contexts C) It is used to polynomial terminology in mathematical analysis D) It replaces all other methods in this domain

Correct: C)

• If you chose A: This is incorrect. Polynomial Terminology serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Polynomial Terminology serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: Polynomial Terminology serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose D: This is incorrect. Polynomial Terminology serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about End Behaviour is TRUE?

A) End Behaviour is an advanced topic beyond this subject's scope B) End Behaviour is mentioned only as a historical footnote C) End Behaviour is a fundamental concept covered in this subject D) End Behaviour is not related to this subject

Correct: C)

• If you chose A: This is incorrect. End Behaviour is a fundamental concept covered in this subject. This subject covers End Behaviour as part of its core content.
• If you chose B: This is incorrect. End Behaviour is a fundamental concept covered in this subject. This subject covers End Behaviour as part of its core content.
• If you chose C: End Behaviour is a fundamental concept covered in this subject. This subject covers End Behaviour as part of its core content. Correct!
• If you chose D: This is incorrect. End Behaviour is a fundamental concept covered in this subject. This subject covers End Behaviour as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) An unrelated numerical value B) A different result from a common mistake C) ** Degree 5, leading coefficient -4 D) The inverse of the correct answer

Correct: C)

• If you chose A: This is incorrect. The worked examples show that the result is ** Degree 5, leading coefficient -4. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is ** Degree 5, leading coefficient -4. The other options represent common errors.
• If you chose C: The worked examples show that the result is ** Degree 5, leading coefficient -4. The other options represent common errors. Correct!
• If you chose D: This is incorrect. The worked examples show that the result is ** Degree 5, leading coefficient -4. The other options represent common errors.

Q5: How are End Behaviour and Roots And Multiplicity related?

A) End Behaviour is the inverse of Roots And Multiplicity B) End Behaviour and Roots And Multiplicity are completely unrelated topics C) End Behaviour is a special case of Roots And Multiplicity D) End Behaviour and Roots And Multiplicity are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both End Behaviour and Roots And Multiplicity are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both End Behaviour and Roots And Multiplicity are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both End Behaviour and Roots And Multiplicity are covered in this subject as interconnected topics.
• If you chose D: Both End Behaviour and Roots And Multiplicity are covered in this subject as interconnected topics. Correct!

Q6: What is a common pitfall when working with Turning Points?

A) The main error with Turning Points is using it when it is not needed B) Turning Points is always computed the same way in all contexts C) A common mistake is confusing Turning Points with a similar concept D) Turning Points has no common misconceptions

Correct: C)

• If you chose A: This is incorrect. Students often confuse Turning Points with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse Turning Points with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: Students often confuse Turning Points with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose D: This is incorrect. Students often confuse Turning Points with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Rational Root Theorem?

A) Use Rational Root Theorem only in pure mathematics contexts B) Rational Root Theorem is not practically useful C) Apply Rational Root Theorem to solve problems in this subject's domain D) Avoid Rational Root Theorem unless explicitly instructed

Correct: C)

• If you chose A: This is incorrect. Rational Root Theorem is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Rational Root Theorem is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: Rational Root Theorem is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose D: This is incorrect. Rational Root Theorem is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Degree and leading coefficient of P(x) = -4x⁵ + 2x³ - x + 7 Answer: Degree 5, leading coefficient -4

2. End behaviour of P(x) = 3x³ - x² + 2 Answer: Degree 3 (odd), leading coefficient 3 (positive). Left end DOWN, right end UP.

3. Multiplicity of root x = 2 in P(x) = (x - 2)⁴(x + 1)² Answer: 4 (even — graph touches and turns)

4. Possible rational roots of P(x) = x³ - 2x² - 5x + 6 Answer: ±1, ±2, ±3, ±6

5. Solve x²(x - 3)(x + 1)² > 0 Answer: Roots at x = 0 (mult 2), x = 3 (mult 1), x = -1 (mult 2). Note x² ≥ 0 always and (x+1)² ≥ 0 always, so sign depends only on (x-3). For x < 3: (x-3) < 0 → product ≤ 0. For x > 3: (x-3) > 0 → product > 0. Solution: (3, ∞).

Summary

Key takeaways:

• Degree = highest power; leading coefficient = coefficient of highest power
• End behaviour: even degree same direction, odd opposite
• Root multiplicity: odd = cross, even = touch
• At most n - 1 turning points for degree n
• Rational Root Theorem: p divides constant, q divides leading coefficient

Pitfalls

• Getting end behaviour wrong: Even-degree polynomials have both ends going the SAME direction (both up or both down, depending on the sign of the leading coefficient). Odd-degree polynomials go OPPOSITE directions. Mixing these up — especially when the leading coefficient is negative — is very common.
• Confusing odd and even multiplicity behaviour: A root with odd multiplicity crosses the x-axis; a root with even multiplicity touches and turns back. Students often remember this backwards or describe touch-behaviour for all multiplicities.
• Forgetting to check all possible rational roots: The Rational Root Theorem gives CANDIDATES, not guarantees. p must divide the constant term, q must divide the leading coefficient. Missing the fractional candidates (like ±1/3 when the leading coefficient is 3) is a frequent oversight.
• Miscounting turning points: A degree n polynomial has at MOST n - 1 turning points, not exactly n - 1. Some polynomials have fewer (e.g., x³ has 0 turning points). Don't force extra turns where they don't belong.
• Assuming every root must be rational: The Rational Root Theorem only finds rational roots. A polynomial can have irrational or complex roots that the theorem will not reveal. If synthetic division doesn't give a zero remainder, the root may be irrational.

Next Steps

Next up: 03-04-rational-functions.md