Class 9 Polynomials: Remainder Theorem, Factor Theorem and Identities

Why this chapter outlasts Class 9

Polynomials in Class 9 is a modest chapter. But factorisation, which it teaches, appears again in quadratics in Class 10, in trigonometric identities, in limits and derivatives in Class 11 and 12, and in every algebra step of a physics or chemistry numerical. Students who find algebra “suddenly hard” in Class 11 often have a Class 9 factorisation gap that was never repaired.

The basics, stated exactly

A polynomial in one variable has whole-number powers only, such as 2x³ − 5x + 7. Know the degree (the highest power), and the names: linear, quadratic, cubic. Know that a zero of a polynomial p(x) is a value of x for which p(x) equals 0, and that a non-zero constant polynomial has no zero. Students often write that a polynomial “has zero” when they mean it has a zero at a particular value. Wording matters here.

Evaluating p(a) is simple substitution, and it is the key tool of the chapter. Insist on putting the value in brackets, especially for negatives: p(−2) with x² becomes (−2)², not −2². A single bracket habit prevents a whole category of lost marks.

Remainder theorem and factor theorem

The remainder theorem says that when p(x) is divided by (x − a), the remainder is p(a). The factor theorem says that (x − a) is a factor of p(x) exactly when p(a) = 0.

1. To find the remainder, do not divide at all: substitute.
2. To test a factor, substitute and see whether the result is zero.
3. To factorise a cubic, guess a root among the factors of the constant term (±1, ±2 and so on), confirm it with the factor theorem, divide to get a quadratic and factorise that.

For questions like “find k so that x − 2 is a factor”, substitute 2, set the result equal to zero, and solve for k. State the theorem in one line before using it; examiners give a mark for it.

Algebraic identities: which to learn first

Learn these cold, in both directions (expanding and factorising):

• (x + y)² = x² + 2xy + y², and the minus version.
• x² − y² = (x + y)(x − y).
• (x + y)³ and (x − y)³, with the 3xy(x + y) term.
• x³ + y³ and x³ − y³, factorised.
• x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx), and its use when x + y + z = 0.

Many students are asked to evaluate 103 × 107 or 99³ without a calculator. The route is an identity applied to a number split cleverly, such as 100 plus or minus a small number. These are good oral exercises for the car or a quiet ten minutes.

Factorisation habits

Teach in this order: take out the common factor first, look for a difference of squares or cubes, then try splitting the middle term for a quadratic. When splitting the middle term, find two numbers whose product is the product of the first and last coefficients and whose sum is the middle one. Finally, multiply the factors back as a check. A student who always checks catches sign errors before the teacher does.

A two-week practice routine

• Week 1: ten minutes daily on expanding and factorising with each identity; evaluate polynomials at positive and negative values.
• Week 2: remainder and factor theorem problems, then cubic factorisation, then a mixed test of fifteen questions in twenty-five minutes.
• Keep an error log: sign, bracket, copying or method. After a fortnight the pattern is usually clear.

Where a tutor helps in Dwarka and Gurugram

A tutor sitting next to the student sees the moment a bracket is dropped, which a marked notebook shows only afterwards. For Class 9 students with some maths anxiety, short daily factorisation drills guided by a tutor often change how the whole subject feels within a month. Home sessions in Dwarka and Gurugram can be fitted around school hours without added travel.

Two worked questions to practise

Find the remainder when p(x) = x³ + 3x² + 3x + 1 is divided by x + 2. Here a = −2, so the remainder is p(−2) = −8 + 12 − 6 + 1 = −1. No long division is needed, which is the whole point of the theorem.

Factorise x³ − 2x² − x + 2. Try x = 1: 1 − 2 − 1 + 2 = 0, so (x − 1) is a factor. Dividing gives x² − x − 2, which factorises as (x − 2)(x + 1). So the full factorisation is (x − 1)(x − 2)(x + 1). Multiply back to check. Notice the sequence the child must follow: guess, test, divide, factorise, check. Learning it as a sequence makes cubics feel like routine work rather than a puzzle.

Questions parents ask

Is this chapter heavily tested in the school exam?

Class 9 papers usually include a mix of short and long polynomial questions, but the pattern depends on the school. The skills are tested again indirectly all year.

My child can factorise but makes sign mistakes.

Use the multiply-back check on every question and practise with negatives deliberately. Most sign errors go away with this one habit.

Should we use a calculator for identities?

No. The point of the exercise is to build the algebra skill. Calculators are not needed at this stage.