📐 Concept diagram

01-09 - Polynomials

Phase: 1 | Subject: 01-09 Prerequisites: 01-08-quadratic-equations.md (quadratic factorising), 01-07-quadratic-expressions.md (expanding brackets) Next subject: 01-10-exponentials-and-logarithms.md

Learning Objectives

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

1. Understand polynomial terminology (degree, coefficient, term)
2. Add and subtract polynomials
3. Multiply polynomials including special products
4. Perform polynomial long division
5. Apply the Remainder Theorem and Factor Theorem

Core Content

Polynomial Terminology

A polynomial is a sum of terms, each being a constant times a variable raised to a non-negative integer power.

Example: 3x⁴ - 2x² + 5x - 7

• Degree: highest exponent (here: 4)
• Leading coefficient: coefficient of highest degree term (here: 3)
• Constant term: term with no variable (here: -7)

Adding and Subtracting Polynomials

Combine LIKE TERMS only.

Common pitfall — sign errors in subtraction: When subtracting polynomials, the minus sign applies to EVERY term in the second polynomial. For example: (3x² + 2x) - (x² - 3x) = 3x² + 2x - x² + 3x, NOT 3x² + 2x - x² - 3x. Think of it as adding the negative: (3x² + 2x) + (-x² + 3x).

Example: (3x² + 2x - 5) + (x² - 3x + 4) = (3x² + x²) + (2x - 3x) + (-5 + 4) = 4x² - x - 1

Example: (2x³ + 4x - 1) - (x³ - 2x + 3) = 2x³ - x³ + 4x + 2x - 1 - 3 = x³ + 6x - 4

Multiplying Polynomials

Use the distributive law (FOIL for binomials).

Example: (x + 3)(x² - 2x + 1) = x(x² - 2x + 1) + 3(x² - 2x + 1) = x³ - 2x² + x + 3x² - 6x + 3 = x³ + x² - 5x + 3

Special Products

(a + b)(a - b) = a² - b² (Difference of squares)

(a + b)² = a² + 2ab + b² (Perfect square)

(a - b)² = a² - 2ab + b² (Perfect square)

Example: (2x + 5)² = 4x² + 20x + 25

Example: (3x - 4)(3x + 4) = 9x² - 16

Polynomial Long Division

Divide polynomials like long division with numbers.

Example: Divide 2x³ + 3x² - x + 5 by x + 1

1. 2x³ ÷ x = 2x². Multiply: 2x²(x + 1) = 2x³ + 2x²
2. Subtract: (2x³ + 3x²) - (2x³ + 2x²) = x²
3. Bring down -x: x² - x
4. x² ÷ x = x. Multiply: x(x + 1) = x² + x
5. Subtract: (x² - x) - (x² + x) = -2x
6. Bring down 5: -2x + 5
7. -2x ÷ x = -2. Multiply: -2(x + 1) = -2x - 2
8. Subtract: (-2x + 5) - (-2x - 2) = 7

Result: 2x³ + 3x² - x + 5 = (x + 1)(2x² + x - 2) + 7

Quotient: 2x² + x - 2, Remainder: 7

Remainder Theorem

When P(x) is divided by (x - a), the remainder is P(a).

Example: Remainder when x³ - 2x² + 4 is divided by (x - 3)? P(3) = 27 - 18 + 4 = 13. Remainder is 13.

Factor Theorem

(x - a) is a factor of P(x) if and only if P(a) = 0.

Example: Is (x - 2) a factor of x³ - 3x² + 4? P(2) = 8 - 12 + 4 = 0. YES, (x - 2) is a factor.

Key Terms

• (a + b)(a - b) = a² - b²
• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• 01 09 Polynomials
• Adding and Subtracting Polynomials
• Correct: A)
• Correct: B)
• Correct: C)
• Example 1: Expand and simplify
• Example 2: Polynomial division
• Example 3: Using the Factor Theorem
• Factor Theorem

Worked Examples

Example 1: Expand and simplify

(2x - 3)(x² + x - 1) = 2x³ + 2x² - 2x - 3x² - 3x + 3 = 2x³ - x² - 5x + 3

Example 2: Polynomial division

Divide x³ + 2x² - 5x + 3 by x - 2.

1. x³ ÷ x = x². Multiply: x²(x - 2) = x³ - 2x²
2. Subtract: (x³ + 2x²) - (x³ - 2x²) = 4x²
3. Bring down -5x: 4x² - 5x
4. 4x² ÷ x = 4x. Multiply: 4x(x - 2) = 4x² - 8x
5. Subtract: (4x² - 5x) - (4x² - 8x) = 3x
6. Bring down 3: 3x + 3
7. 3x ÷ x = 3. Multiply: 3(x - 2) = 3x - 6
8. Subtract: (3x + 3) - (3x - 6) = 9

Result: x³ + 2x² - 5x + 3 = (x - 2)(x² + 4x + 3) + 9

Example 3: Using the Factor Theorem

Determine whether (x + 2) is a factor of P(x) = x³ + 3x² - 4x - 12, and factorise completely if possible.

1. Check if P(-2) = 0: P(-2) = (-2)³ + 3(-2)² - 4(-2) - 12
2. = -8 + 3(4) + 8 - 12
3. = -8 + 12 + 8 - 12 = 0 ✓
4. Since P(-2) = 0, (x + 2) is a factor by the Factor Theorem.
5. Divide P(x) by (x + 2) to find the other factor:
6. x³ ÷ x = x². Multiply: x²(x + 2) = x³ + 2x²
7. Subtract: (x³ + 3x²) - (x³ + 2x²) = x²
8. Bring down -4x: x² - 4x
9. x² ÷ x = x. Multiply: x(x + 2) = x² + 2x
10. Subtract: (x² - 4x) - (x² + 2x) = -6x
11. Bring down -12: -6x - 12
12. -6x ÷ x = -6. Multiply: -6(x + 2) = -6x - 12
13. Subtract: 0 (remainder 0 confirms factor)
14. Quotient: x² + x - 6
15. Factorise the quotient: x² + x - 6 = (x + 3)(x - 2)
16. Complete factorisation: P(x) = (x + 2)(x + 3)(x - 2)

This shows how the Factor Theorem reduces the work: find one root, divide, then factorise the simpler quotient.

Quiz

Q1: What does the concept of Adding and Subtracting Polynomials primarily refer to in this subject?

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

Correct: B)

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

Q2: What is the primary purpose of Factor Theorem?

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

Correct: B)

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

Q3: Which statement about Polynomial Terminology is TRUE?

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

Correct: D)

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

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

A) The inverse of the correct answer B) An unrelated numerical value C) A different result from a common mistake D) (x + 1)(2x² + x - 2) + 7

Correct: D)

• If you chose A: This is incorrect. The worked examples show that the result is (x + 1)(2x² + x - 2) + 7. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is (x + 1)(2x² + x - 2) + 7. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is (x + 1)(2x² + x - 2) + 7. The other options represent common errors.
• If you chose D: The worked examples show that the result is (x + 1)(2x² + x - 2) + 7. The other options represent common errors. Correct!

Q5: How are Polynomial Terminology and Multiplying Polynomials related?

A) Polynomial Terminology and Multiplying Polynomials are closely related concepts B) Polynomial Terminology is a special case of Multiplying Polynomials C) Polynomial Terminology is the inverse of Multiplying Polynomials D) Polynomial Terminology and Multiplying Polynomials are completely unrelated topics

Correct: A)

• If you chose A: Both Polynomial Terminology and Multiplying Polynomials are covered in this subject as interconnected topics. Correct!
• If you chose B: This is incorrect. Both Polynomial Terminology and Multiplying Polynomials are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Polynomial Terminology and Multiplying Polynomials are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both Polynomial Terminology and Multiplying Polynomials are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Special Products?

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

Correct: B)

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

Q7: When should you apply Polynomial Long Division?

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

Correct: D)

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

Practice Problems

1. Add: (3x² - 2x + 1) + (x² + 4x - 5)
   Click for answer

4x² + 2x - 4

1. Multiply: (x + 2)(x - 3)
   Click for answer

x² - x - 6

1. Expand: (2x - 1)²
   Click for answer

4x² - 4x + 1

1. Divide (x² + 5x + 6) by (x + 2)
   Click for answer

Quotient: x + 3, Remainder: 0. (x + 2) is a factor.

1. Remainder when P(x) = x³ - 4x + 2 is divided by (x - 1)
   Click for answer

P(1) = 1 - 4 + 2 = -1

1. Is (x + 3) a factor of x² + 2x - 15?
   Click for answer

P(-3) = 9 - 6 - 15 = -12 ≠ 0. No, not a factor. (x + 5) and (x - 3) are the factors.

Summary

Key takeaways:

• Polynomial: sum of terms with non-negative integer exponents
• Degree = highest exponent; leading coefficient = coefficient of highest term
• Add/subtract: combine like terms
• Multiply: distribute every term (FOIL for binomials)
• Special products: (a+b)², (a-b)², (a+b)(a-b) = a²-b²
• Long division: divide, multiply, subtract, repeat
• Remainder Theorem: P(a) = remainder when dividing by (x - a)
• Factor Theorem: (x - a) is a factor iff P(a) = 0

Pitfalls

• Sign errors when subtracting polynomials. When subtracting (x³ - 2x + 3) from (2x³ + 4x - 1), the minus sign applies to EVERY term: 2x³ + 4x - 1 - x³ + 2x - 3. A common mistake is writing -x³ - 2x + 3, forgetting that subtracting -2x gives +2x and subtracting +3 gives -3.
• Confusing degree with leading coefficient. The degree is the highest exponent (e.g., 5 in 3x⁵ - x² + 7), while the leading coefficient is the number multiplying that term (3). These are distinct concepts.
• Forgetting to bring down terms during polynomial long division. After each subtraction step, you must bring down the NEXT term before continuing. Skipping this produces a wrong quotient. Follow the pattern: divide, multiply, subtract, bring down, repeat.
• Misapplying the Factor Theorem with sign of the root. To check if (x + 2) is a factor, evaluate P(-2), not P(2). The factor (x - a) corresponds to evaluating P(a). So (x + 2) = (x - (-2)), meaning test a = -2.
• Assuming the Remainder Theorem gives the quotient. P(a) gives only the REMAINDER when dividing by (x - a), not the quotient. To find the quotient, you must perform the division. The theorem is a shortcut for the remainder only.

Next Steps

Next up: 01-10-exponentials-and-logarithms.md