CAT Quant · Study & Practice

Polynomials

AreaAlgebra DifficultyModerate CAT weightage1–3 questions (directly + inside equations, functions, maxima–minima, number system)

Polynomials sit quietly behind a large slice of CAT Algebra. A polynomial is just an expression built from a variable using addition, subtraction and whole-number powers — and almost every quadratic, cubic, identity-based simplification or roots question is really a polynomials question wearing a different label. CAT rarely asks you to "expand (a+b)³" in the open; instead it hides the algebraic identities inside ugly-looking arithmetic (compute 103³ in your head), buries the factor theorem inside a divisibility problem, or asks for the sum of cubes of the roots of a cubic without ever solving it. The students who score here are the ones who recognise the standard forms instantly and reach for the right tool — an identity, the remainder theorem, the factor theorem, or Vieta’s relations — instead of grinding. This chapter builds that pattern-recognition: the core identities for squares and cubes, the remainder and factor theorems for division, and the symmetric-function relations that connect the coefficients of a polynomial to its roots. Each topic carries worked CAT-style examples, the fastest line of attack, and the traps that quietly cost marks.

Topics

1 Polynomial Identities 4 worked · 5 practice · 8-Q test Start → 2 Factor Theorem 4 worked · 5 practice · 8-Q test Start → 3 Remainder Theorem 4 worked · 5 practice · 8-Q test Start → 4 Roots of Polynomials 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Read ugly numbers as identities: 103² = (100+3)², 97×103 = (100−3)(100+3) = 10000 − 9.
  • x + 1/x given ⇒ x² + 1/x² = (x + 1/x)² − 2 and x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x).
  • a + b + c = 0 is a gift: it forces a³ + b³ + c³ = 3abc instantly.
  • Factor theorem: to factor a cubic, test divisors of the constant term; (x − 1) works when coefficients sum to 0.
  • Remainder of P(x) ÷ (x − a) is just P(a) — never do long division in the exam.
  • For roots, use Vieta first: Σα = −b/a, Σαβ = c/a, αβγ = −d/a; then Σα² = (Σα)² − 2Σαβ.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Dropping the middle term: (a + b)² is a² + 2ab + b², not a² + b².
  • Sign slips in Vieta — forgetting product of roots of a cubic is −d/a, not d/a.
  • Using x = a (not x = −a) when dividing by (x + a) in the remainder theorem.
  • Misremembering a³ − b³ = (a − b)(a² + ab + b²): the middle sign is +, opposite to the binomial’s.
  • Assuming a + b + c = 0 makes a³ + b³ + c³ = 0; it equals 3abc, not 0.

📈 CAT exam insight & PYQ analysis

In CAT, pure polynomial questions are uncommon but the underlying tools surface constantly. Algebraic identities show up disguised as arithmetic (evaluate a large square or product fast) and inside x + 1/x manipulations that feed maxima–minima and surds problems. The remainder theorem appears wherever divisibility meets a base-as-variable trick, and Vieta’s relations are tested through "find Σα² or Σ1/α without solving" in quadratics and cubics. XAT and SNAP lean a little more on direct factorisation and factor-theorem set-ups. Prioritise instant recall of the cube identities, the three-cube collapse to 3abc, and clean sign handling in Vieta — those three carry most of the marks.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

(a + b)² = ?Tap to reveal
a² + 2ab + b²
(a − b)³ = ?Tap to reveal
a³ − 3a²b + 3ab² − b³
a³ + b³ factorised?Tap to reveal
(a + b)(a² − ab + b²)
a³ − b³ factorised?Tap to reveal
(a − b)(a² + ab + b²)
a³ + b³ + c³ − 3abc = ?Tap to reveal
(a + b + c)(a² + b² + c² − ab − bc − ca)
If a + b + c = 0, then a³ + b³ + c³ = ?Tap to reveal
3abc
Remainder of P(x) ÷ (x − a)?Tap to reveal
P(a)
Factor theorem condition?Tap to reveal
(x − a) divides P(x) ⇔ P(a) = 0
Quadratic ax²+bx+c: sum and product of roots?Tap to reveal
Sum = −b/a, product = c/a
Cubic ax³+bx²+cx+d: αβγ = ?Tap to reveal
−d/a
α² + β² + γ² in terms of Vieta sums?Tap to reveal
(Σα)² − 2Σαβ
x³ + 1/x³ from x + 1/x = s?Tap to reveal
s³ − 3s

📌 Quick revision

Polynomials reduce to a small toolkit. Memorise the identities: (a ± b)², (a ± b)³, a² − b², a³ ± b³, and the three-cube form a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca), which becomes 3abc when a+b+c = 0. Use x+1/x relations to climb to higher powers. The remainder of P(x) ÷ (x − a) is P(a); (x − a) is a factor exactly when P(a) = 0. For roots, lean on Vieta: Σα = −b/a, Σαβ = c/a, αβγ = −d/a, then Σα² = (Σα)² − 2Σαβ. Watch signs in −b/a and −d/a, and never confuse the middle-term sign in a³ − b³.

Chapter test

📝 Polynomials — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Polynomials when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards