📌 Key Points
- Polynomial: algebraic expression with non-negative integer exponents and real coefficients
- Degree: highest exponent of x. Degree 0 = constant, 1 = linear, 2 = quadratic, 3 = cubic
- Zero of p(x): value 'a' such that p(a) = 0. Geometrically, x-intercepts.
- Polynomial of degree n has at most n real zeros
- If 'a' is a zero, then (x - a) is a factor of p(x)
- Sum of zeros (quadratic): α + β = -b/a
- Product of zeros (quadratic): αβ = c/a
- Division Algorithm: p(x) = g(x)×q(x) + r(x), where degree(r) < degree(g)
- Remainder Theorem: Remainder when p(x) divided by (x-a) is p(a)
- Factor Theorem: (x-a) is a factor of p(x) ⟺ p(a) = 0
- For cubic with zeros α, β, γ: Sum = -b/a, Sum of products (pairs) = c/a, Product = -d/a
- Factorization methods: common factors, AC method, algebraic identities, factor theorem
- AC method: For ax² + bx + c, find factors of ac that sum to b
- If (x - a) is a factor, we can divide p(x) by (x - a) to find quotient polynomial
- To find zeros of cubic: test rational roots using factor theorem, then factor remaining quadratic
- Forming polynomial from zeros: p(x) = a(x - α)(x - β)(x - γ)...
- Algebraic identities: a² - b² = (a-b)(a+b); a³ ± b³ = (a ± b)(a² ∓ ab + b²)
- Remainder Theorem + Division: p(x) = (x - a)×q(x) + p(a)
- To verify factorization: expand and check if product equals original polynomial
- Complex roots: non-real zeros come in conjugate pairs for polynomials with real coefficients
📘 Important Definitions
🔢 Formulas & Laws
Quadratic Polynomial Form
ax² + bx + c
General form of degree 2 polynomial
Sum of Zeros (Quadratic)
α + β = -b/a
For p(x) = ax² + bx + c
Product of Zeros (Quadratic)
αβ = c/a
For p(x) = ax² + bx + c
Forming Quadratic from Zeros
p(x) = x² - (sum of zeros)x + (product of zeros)
If zeros are α and β: p(x) = x² - (α+β)x + αβ
Division Algorithm
p(x) = g(x) × q(x) + r(x)
degree(r) < degree(g)
Remainder Theorem
Remainder = p(a) when p(x) is divided by (x - a)
Quick way to find remainder without division
Factor Theorem
(x - a) is a factor ⟺ p(a) = 0
Most important theorem in polynomial analysis
Cubic Polynomial Form
ax³ + bx² + cx + d
General form of degree 3 polynomial
Sum of Zeros (Cubic)
α + β + γ = -b/a
For p(x) = ax³ + bx² + cx + d
Sum of Products of Zeros (Cubic)
αβ + βγ + γα = c/a
Products taken two at a time
Product of Zeros (Cubic)
αβγ = -d/a
For p(x) = ax³ + bx² + cx + d
Difference of Squares
a² - b² = (a - b)(a + b)
Used for factorization
⚠️ Common Mistakes
✗ Wrong: Confusing polynomial with non-polynomial (e.g., √x or 1/x are NOT polynomials)
✓ Correct: Polynomials must have non-negative integer exponents only
✗ Wrong: Wrong sign in sum of zeros formula: using α + β = b/a instead of -b/a
✓ Correct: Sum of zeros = -b/a (note the negative sign)
✗ Wrong: Not verifying zeros by substitution
✓ Correct: Always check: p(a) = 0 to confirm 'a' is a zero
✗ Wrong: Incomplete factorization of polynomials
✓ Correct: Factor completely until all factors are irreducible
✗ Wrong: Confusing remainder theorem with factor theorem
✓ Correct: Remainder = p(a) always; factor exists only if p(a) = 0
✗ Wrong: Incorrect application of AC method
✓ Correct: Find factors of ac (not just a or c) that sum to b
✗ Wrong: Forgetting constraint: degree of remainder < degree of divisor
✓ Correct: In p(x) = g(x)q(x) + r(x), remainder must have lower degree than divisor
✗ Wrong: Assuming polynomial has no real zeros just because factorization isn't obvious
✓ Correct: Test rational roots using factor theorem systematically
📝 Exam Focus
These questions are frequently asked in CBSE exams:
🎯 Last-Minute Recall
Close your eyes and try to recall: Key definitions, formulas, and 3 common mistakes. If you can recall 80% without looking, you're exam-ready!