Polynomials Class 9 Notes: Degree, Zeros, Remainder Theorem and Factor Theorem

Polynomials (Class 9 Maths, Chapter 2) builds the algebra you use till Class 12. The chapter is short but the remainder theorem, factor theorem and identities appear in exams every year.

Basic terms

• Polynomial in x: an expression like 3x² + 2x − 5 with whole-number powers only. (x + 1/x is not a polynomial.)
• Degree: highest power of x. Linear (1), quadratic (2), cubic (3).
• Zero of a polynomial: a value k such that p(k) = 0. For 2x − 6, zero is 3.

Remainder theorem

If p(x) is divided by (x − a), the remainder is p(a).

Example: Remainder when x³ − 3x² + 4x + 5 is divided by x − 2: p(2) = 8 − 12 + 8 + 5 = 9. No long division needed.

Factor theorem

(x − a) is a factor of p(x) if and only if p(a) = 0.

Example: Is x + 1 a factor of x³ + x² + x + 1? Put x = −1: −1 + 1 − 1 + 1 = 0. Yes.

Factorising a cubic (exam favourite)

Factorise x³ − 6x² + 11x − 6.

1. Try factors of the constant term 6: ±1, ±2, ±3, ±6. p(1) = 1 − 6 + 11 − 6 = 0 → (x − 1) is a factor.
2. Divide: x³ − 6x² + 11x − 6 = (x − 1)(x² − 5x + 6).
3. Factorise the quadratic: (x − 2)(x − 3).

Answer: (x − 1)(x − 2)(x − 3).

Algebraic identities you must memorise

• (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²
• a² − b² = (a + b)(a − b)
• (x + a)(x + b) = x² + (a + b)x + ab
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a + b)³ = a³ + b³ + 3ab(a + b); (a − b)³ = a³ − b³ − 3ab(a − b)
• a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²)
• a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca); if a + b + c = 0 then a³ + b³ + c³ = 3abc

Trick question: Evaluate 104 × 96 = (100 + 4)(100 − 4) = 10000 − 16 = 9984.

Common mistakes

• Using p(−a) instead of p(a) for divisor (x − a). For x + 2, put x = −2.
• Forgetting the 3ab(a + b) term in cube identities.

Next chapter link: Quadratic Equations Class 10.

Stuck on a polynomial or identity question?

Type the exact question from your book, pick your class and language (Hindi, Hinglish, English or 9 more), and ask your doubt on DoubtPay AI. You get step-by-step working with a drawn diagram where it helps. No subscription — credit packs from ₹9, and unused credits stay in your account.

Frequently asked questions

What is the remainder theorem in simple words?

To find the remainder when dividing by x − a, just put x = a in the polynomial.

How many zeros can a cubic polynomial have?

At most 3.

Is 5 a polynomial?

Yes, a constant polynomial of degree 0.