📚Concepts

📚 Polynomial: An algebraic expression like 𝟑𝑥² + 𝟓𝑥 − 𝟐 made up of terms with variables and powers.

📚 Degree of a Polynomial: The highest power of the variable in the polynomial.

• 𝐸𝑥𝑎𝑚𝑝𝑙𝑒: Degree of 𝟒𝑥³ + 𝟐𝑥² + 𝟓 is 𝟑.

📚 Zero of a Polynomial: The value of 𝑥 where 𝑓(𝑥) = 𝟎. It is also called a root or solution.

📚 Types of Polynomials:

🔸 Constant: Only number

(e.g., 𝟕)

🔹 Linear: Degree 𝟏

(e.g., 𝟐𝑥 + 𝟑)

🔸 Quadratic: Degree 𝟐

(e.g., 𝑥² − 𝟒𝑥 + 𝟑)

🔹 Cubic: Degree 𝟑

(e.g., 𝑥³ − 𝑥² + 𝑥 − 𝟏)

📚 Factorisation: Breaking a polynomial into simpler expressions called factors.

• 𝐸𝑥𝑎𝑚𝑝𝑙𝑒: 𝑥² − 𝟒 = (𝑥 − 𝟐) (𝑥 + 𝟐)

📚 Remainder Theorem: If a polynomial 𝑓(𝑥) is divided by (𝑥 − 𝑎), the remainder = 𝑓(𝑎).

📚 Factor Theorem: If 𝑓(𝑎) = 𝟎, then (𝑥 − 𝑎) is a factor of 𝑓(𝑥).

✍️ Formulas

✍️ Standard Polynomial Form:  

𝑓(𝑥) = 𝑎ₙ𝑥ⁿ + 𝑎ₙ₋₁𝑥ⁿ⁻¹ + ⋯ + 𝑎₁𝑥 + 𝑎₀ → Where 𝑎ₙ ≠ 𝟎 and n is a non-negative integer

✍️ Remainder Theorem:  

If 𝑓(𝑥) is divided by (𝑥 − 𝑎), then Remainder = 𝑓(𝑎)

✍️ Factor Theorem:

If 𝑓(𝑎) = 𝟎, then (𝑥 − 𝑎) is a factor of 𝑓(𝑥)

✍️ Relationship between Zeros and Coefficients (Quadratic Polynomial):

For 𝑓(𝑥) = 𝑎𝑥² + 𝑏𝑥 + 𝑐, let 𝛼 and 𝛽 be the zeros, then:  

• 𝛼 + 𝛽 = −𝑏⁄𝑎    

• 𝛼 ⋅ 𝛽 = 𝑐⁄𝑎

✍️ Number of Zeros and Degree:  

Every polynomial of degree 𝑛 can have at most 𝑛 zeros

✍️ Types of Polynomials by Degree:  

• Degree 𝟏 → Linear 

  (e.g., 𝟐𝑥 + 𝟑)  

• Degree 𝟐 → Quadratic 

  (e.g., 𝑥² − 𝟒)  

• Degree 𝟑 → Cubic

  (e.g., 𝑥³ + 𝟐𝑥 + 𝟏)