IPMAT Polynomials Questions 2027: Solved Examples with Solutions

Author : Lalita Vishwakarma

October 13, 2026

If you've ever looked at a quartic in a mock and just sort of skipped it on the basis of habit, then this page is for you. The IPMAT polynomial questions can look absolutely terrifying because of the high powers involved, but almost all of them are, in fact, just a simple quadratic once you spot the trick.

This guide is for Class 12 students and droppers preparing for IPMAT Indore and Rohtak and for Class 11 students who want to get into algebra early. You'll get the key formulas on one page, IPMAT polynomial MCQs with a hint and answer for each, the common errors that cost students marks and a quick plan to get the topic down. Do each question before you open the hint - that one habit is what turns reading into actual practice.

What are IPMAT Polynomial Questions?

IPMAT polynomial questions test your ability to find roots, factorise expressions, use the sum and product of roots and apply the remainder theorem. They are part of the Quantitative Ability (maths) section. IIM Indore have had at least one polynomial question every year from 2023 to 2026 (3 or more in 2025 and 2026). Have a go at the IPMAT polynomial questions with solutions below to see precisely how the topic is being tested.

Key Highlights

Polynomials now show up more often in IPMAT QA: 1 tagged question in 2023, 2 in 2024, 4 in 2025 and 3 in 2026.
The vast majority of IPMAT polynomial questions are solved using one of four techniques: sum and product of roots, the remainder/factor theorem, substitution or graph sketching.
They're available as MCQs (+4 for a correct answer, -1 for a wrong one) and as short answer questions (no penalty for a wrong answer), so again it's all about getting the answer right.
The same themes come up every year; just the specific questions vary. Learn the methodology, not the answer.
Three weeks of focused practice can turn polynomials from a weak topic into a scoring one.

IPMAT Polynomials Questions with Solutions to Practice Now

Practice the important polynomials questions for IPMAT 2027 with solutions:

Q1. The equation 2ˣ − x² = 0 has:

(A) exactly one real solution

(B) exactly two real solutions

Hint: Don't solve algebraically — sketch y = 2ˣ and y = x² and count the crossings, including for negative x.

Correct answer: (D)

Explanation

x = 2 and x = 4 work directly. For a negative root: at x = −1, 2ˣ = 0.5 is less than x² = 1; at x = 0, 2ˣ = 1 is more than 0. The curves cross between −1 and 0, giving three real solutions. Most students stop at two.

Q2. The number of values a can take such that x⁴ + ax³ + (3a − 4)x² + 2(a − 1)x − 4 can be written as (x² + px + 2)(x² + qx − 2), where p and q are real, is:

(A) 2

(B) 1

(D) 3

Hint: Expand the product and compare coefficients of x³, x² and x. You get three equations in p, q and a.

Correct answer: (B)

Explanation

Expanding gives x³: p + q = a; x²: pq = 3a − 4; x: 2(q − p) = 2(a − 1). From the first and third, p = 1/2 and q = a − 1/2. Then pq = (2a − 1)/4 = 3a − 4, so a = 1.5. Only one value.

Q3. If x² − 12x + 35 = 0 and x² + ax + 105 = 0 have at least one common root. For real y, what will be the minimum possible value of y² + 4y − 5a is:

(A) 96

(B) 106

(D) 116

Hint: Find the roots of the first equation, then use product of roots = 105 to get the possible values of a.

Correct answer: (B) 106

Explanation

The first equation has roots 5 and 7. If 5 is common, the other root is 21 and a = −26; if 7 is common, the other is 15 and a = −22. The minimum of y² + 4y is −4 (at y = −2), so the expression is −4 − 5a. Its minimum uses the larger a = −22: −4 + 110 = 106. Option (C) is what you get if you wrongly use a = −26.

Q4. If a₁, a₂, …, a₈ are the roots of x⁸ + x⁷ + … + x + 1 = 0, then what will be the correct answer for a₁²⁰²⁵ + a₂²⁰²⁵ + … + a₈²⁰²⁵ :

(A) 0

(B) 2

(D) 4

Hint: Multiply the left side by (x − 1). What equation do the roots now satisfy?

Correct answer: (C) 8

Explanation: (x − 1)(x⁸ + … + 1) = x⁹ − 1, so every root satisfies a⁹ = 1. Since 2025 = 9 × 225, each a²⁰²⁵ = 1, and the sum of eight such terms is 8.

Q5. In 8x² − 2kx + k = 0, one root is p times the other, where p and k are positive real numbers. Then k equals:

(A) p + 1/p (B) 2(p + 1/p) (C) 2(√p + 1/√p)² (D) (√p + 1/√p)²

Hint: When one root is p times the other, this shortcut always works: (p + 1)²/p = b²/(ac).

Correct answer: (C)

Explanation: Here b²/(ac) = 4k²/(8k) = k/2. So k = 2(p + 1)²/p. Since (p + 1)²/p = p + 2 + 1/p = (√p + 1/√p)², k = 2(√p + 1/√p)².

Q6. P(x) is a quadratic with P(0) = 2, P(1) + P(2) + P(3) = 14, and the determinant |P(0) P(1); P(0) P(2)| = 0. Then P(4) equals:

(A) −14

(B) 30

(D) 16

Hint: Expand the determinant first. Because P(0) ≠ 0, it tells you something simple about P(1) and P(2).

Correct answer: (C) −6

Explanation: The determinant is P(0)[P(2) − P(1)] = 0, so P(1) = P(2). Write P(x) = ax² + bx + 2. P(1) = P(2) gives b = −3a. The sum condition gives 14a + 6b + 6 = 14, so −4a = 8, a = −2, b = 6. P(4) = −32 + 24 + 2 = −6.

Q7. f(x) = a²x² + 2bx + c, where a ≠ 0 and b, c are real. Then:

(A) f(x) has a maximum and a minimum

(B) f(x) has a minimum and no maximum

(D) f(x) has no minimum and no maximum

Hint: Look only at the sign of the x² coefficient.

Correct answer: (B)

Explanation: a² > 0 for any non-zero a, so the parabola opens upward. It has a lowest point and rises without limit — a minimum, no maximum.

Q8. The number of real solutions of (x² − 15x + 55)^(x² − 5x + 6) = 1 is:

(A) 4

(B) 5

(D) 2

Hint: bᵉ = 1 in three cases: base = 1; exponent = 0 with base ≠ 0; base = −1 with an even exponent.

Correct answer: (C) 6

Explanation: Base = 1: x² − 15x + 54 = 0, so x = 6, 9. Exponent = 0: x = 2, 3 (base is 29 and 19, both non-zero). Base = −1: x² − 15x + 56 = 0, so x = 7, 8, with exponents 20 and 30, both even. Total: 6. Option (A) is what you get if you forget the −1 case.

Q9. The difference between the maximum and minimum real roots of (x² − 5)⁴ + (x² − 7)⁴ = 16 is:

(A) √10

(B) 2√5

(D) 2√7

Hint: Let t = x² − 6, the average of the two brackets.

Correct answer: (D) 2√7.

Explanation: The equation becomes (t + 1)⁴ + (t − 1)⁴ = 16, which simplifies to 2t⁴ + 12t² + 2 = 16, or t⁴ + 6t² − 7 = 0. So t² = 1, t = ±1, giving x² = 7 or 5. The largest root is √7 and the smallest −√7; the difference is 2√7.

Q10.The harmonic mean of the roots of (5 + √2)x² − bx + 8 + 2√5 = 0 is 4. The value of b is:

(A) 2

(B) 4 − √5

(D) 4 + √5

Hint: HM of two roots = 2αβ/(α + β). Write it in terms of the coefficients — the leading coefficient cancels.

Correct answer: (D) 4 + √5

Explanation: HM = 2(c/a)/(b/a) = 2c/b = 4, so b = c/2 = (8 + 2√5)/2 = 4 + √5. You never need the (5 + √2) — it's only there to confuse you.

Q11. If α and β are the roots of x² − 5x + 3 = 0, then α³ + β³ equals:

(A) 70

(B) 80

(D) 125

Hint: Never find the roots. Use α³ + β³ = (α + β)³ − 3αβ(α + β).

Correct answer: (B) 80

Explanation: α + β = 5 and αβ = 3, so 125 − 3 × 3 × 5 = 125 − 45 = 80.

Q12. (Practice · Medium) The remainder when x¹⁰⁰ is divided by x² − 3x + 2 is:

(A) (2¹⁰⁰ − 1)x + 2 − 2¹⁰⁰ (B) (2¹⁰⁰ + 1)x − 2¹⁰⁰ (C) 2¹⁰⁰x − 1 (D) (2¹⁰⁰ − 1)x + 1

Hint: Dividing by a quadratic leaves a linear remainder rx + s. Plug in the roots of the divisor, x = 1 and x = 2.

Correct answer: (A)

Explanation: Write x¹⁰⁰ = Q(x)(x − 1)(x − 2) + rx + s. At x = 1: r + s = 1. At x = 2: 2r + s = 2¹⁰⁰. So r = 2¹⁰⁰ − 1 and s = 2 − 2¹⁰⁰. Quick check: option (A) at x = 1 gives 1, as it must.

Q13. (Practice · Medium) If x³ + ax² − bx + 10 is divisible by x² − 3x + 2, then a + b equals:

(A) 9

(B) 11

(D) 15

Hint: x² − 3x + 2 = (x − 1)(x − 2). By the factor theorem, the polynomial is 0 at both x = 1 and x = 2.

Correct answer: (D) 15

Explanation: At x = 1: a − b = −11. At x = 2: 4a − 2b = −18, so 2a − b = −9. Subtracting gives a = 2, then b = 13. a + b = 15.

Q14. (Practice · Medium) P(x) is a cubic polynomial with leading coefficient 1, and P(1) = 1, P(2) = 2, P(3) = 3. Then P(4) equals:

(A) 4

(B) 6

(D) 12

Hint: Define Q(x) = P(x) − x. What are the roots of Q(x)?

Correct answer: (C) 10

Explanation: Q(x) is a monic cubic that is zero at 1, 2 and 3, so Q(x) = (x − 1)(x − 2)(x − 3). Then P(4) = Q(4) + 4 = 6 + 4 = 10. Students who answer 4 assume P(x) = x, which isn't a cubic.

Q15. (Practice · Easy) The sum of the squares of the roots of x³ − 6x² + 11x − 6 = 0 is:

(A) 14

(B) 22

(D) 36

Hint: Σα² = (Σα)² − 2Σαβ. Read both sums straight from the coefficients.

Correct answer: (A) 14

Explanation: Σα = 6 and Σαβ = 11, so 36 − 22 = 14. Check: the roots are 1, 2 and 3, and 1 + 4 + 9 = 14.

IPMAT Polynomials Practice Questions Answer Key

Q	Answer	Q	Answer	Q	Answer
1	D	6	C	11	B
2	B	7	B	12	A
3	B	8	C	13	D
4	C	9	D	14	C
5	C	10	D	15	A

How Important Are Polynomial Questions for IPMAT?

Polynomials can be a high-value topic to prepare for IPMAT Indore. In recent years, IIM Indore has featured 1–4 questions on polynomials in the Quantitative Ability section, making this topic worth your attention.

With each question carrying 4 marks, that’s up to 16 marks potentially up for grabs from a single topic!

Polynomials: Year-Wise Question Trend

Year	Number of Questions
2023	1
2024	2
2025	4
2026	3

The takeaway: Don’t overlook polynomials in your IPMAT preparation. With questions appearing every year in this period, mastering the key concepts and practising previous-year questions could help you secure valuable marks.

Where polynomials sit in the IPMAT Indore pattern

IPMAT Indore 2026 had three 40-minute sections

Section	Questions	Marking	Polynomials appear?
Quantitative Ability (MCQ)	30	+4 / −1	Yes — most PYQs are here
Quantitative Ability (Short Answer)	15	+4 / 0	Yes — 2024 and 2026 had one each
Verbal Ability (MCQ)	45	+4 / −1	No

IIM Rohtak's IPMAT has a standalone QA section with only MCQs. Algebra is part of that syllabus, so the same root-based skills will be required for Rohtak's IPMAT QA questions.

IPMAT Polynomial Questions by Year (IIM Indore)

Year	Tagged polynomials PYQs	What was tested
2026	3 (2 MCQ + 1 SA)	Graph intersections, factorising a quartic, common roots
2025	4 (MCQ)	Roots of unity, ratio of roots, quadratic from conditions, max/min
2024	2 (1 MCQ + 1 SA)	Exponential equations, biquadratic substitution
2023	1 (MCQ)	Harmonic mean of roots

Quadratic equations and functions are counted separately, so algebra as a whole has even more questions.

What this tells you: IIM Indore has moved from one simple formula question to longer questions that mix polynomials with graphs, powers and complex roots. Memorising formulas is not enough. You need to see the simple idea hiding inside the question.

Polynomials Formula Sheet for IPMAT 2027 Exam

Six ideas solve almost every IPMAT polynomials question. Learn when to use each one, not just the formula.

A polynomial is an expression made of terms of the form a·xⁿ, where n is a whole number (0, 1, 2, …). Its degree is the highest power of x. A degree-n polynomial has exactly n roots when complex roots are counted.

Concept	Rule	Use it when the question says…
Sum and product of roots (Vieta)	For ax² + bx + c: α + β = −b/a, αβ = c/a	"roots are in a ratio", "HM/AM of roots", "α³ + β³"
Vieta for a cubic	For ax³ + bx² + cx + d: Σα = −b/a, Σαβ = c/a, αβγ = −d/a	"sum of squares of roots" of a cubic
Remainder theorem	Remainder of P(x) ÷ (x − k) is P(k)	"find the remainder"
Factor theorem	(x − k) is a factor if P(k) = 0	"divisible by", "is a factor of"
Discriminant	D = b² − 4ac; D > 0 two real roots, D = 0 equal, D < 0 none	"real roots", "number of solutions"
Roots of unity	Roots of xⁿ − 1 = 0 satisfy ωⁿ = 1	"1 + x + x² + … + xⁿ⁻¹ = 0"

Common Mistakes in IPMAT Polynomial Questions for 2027

Polynomials might look like a straightforward topic, but a small mistake can cost you an easy 4 marks in IPMAT. Many students understand the concepts yet lose marks because they rush through calculations, miss hidden conditions, or apply the wrong formula.

If you’re preparing for IPMAT 2027, here are some common mistakes to watch out for.

1. Memorising Formulas Without Understanding Them

Knowing identities like (a^2-b^2=(a-b)(a+b)) is useful, but memorising formulas alone won’t get you very far. IPMAT questions can test whether you know when and how to apply them.

What to do: Practise questions that use the same identity in different ways. Learn to recognise patterns instead of relying on a fixed approach.

2. Ignoring the Relationship Between Roots and Coefficients

Questions involving the roots of a quadratic polynomial often use relationships such as:

Sum of roots: (\alpha+\beta=-\frac{b}{a})
Product of roots: (\alpha\beta=\frac{c}{a})

A common mistake is forgetting the negative sign in the sum of roots or mixing up the sum and product.

What to do: Write down the standard relationships and solve questions involving expressions such as (\alpha^2+\beta^2), (\frac{1}{\alpha}+\frac{1}{\beta}), and (\alpha^3+\beta^3).

3. Forgetting Special Cases

Students sometimes assume that every polynomial question has one standard solution. However, questions may involve equal roots, repeated factors, or restrictions on the possible values of a variable..

4. Making Sign Errors While Factorising

Factorisation is often where students lose marks unnecessarily. A negative sign in the wrong place can lead to incorrect roots and a completely wrong answer.

What to do: After factorising, multiply the factors back together to verify that you get the original polynomial.

5. Spending Too Much Time on One Question

In IPMAT, knowing the concept is only half the battle. You also need to manage your time. Spending several minutes on one difficult polynomial question can mean missing easier questions elsewhere.

What to do: If you cannot identify a clear approach after your initial attempt, move on and return to the question later. Practise timed sets to improve your speed and accuracy.

6. Skipping Previous-Year Questions

Textbook exercises help build your basics, but previous-year questions give you a better idea of how concepts can be tested in the actual exam.

What to do: Solve IPMAT Indore previous-year questions on polynomials and maintain a record of the questions you get wrong. Revisit them after a few days to check whether you have understood the underlying concept.

How to Prepare Polynomial Questions for IPMAT 2027?

Think of polynomials as a "spot the pattern" topic: the same six ideas come back every year, just written differently.

Mentor's strategy

Read the ask before the equation. If it asks for a sum, a product or something like α² + β², you usually don't need the roots at all.
Count cases out loud. "How many solutions" questions reward students who write every case before solving any.
Plug in to eliminate. In MCQs, put x = 0, 1 or a root into the options. Q12 can be solved in 20 seconds this way.
Keep a trap log. After each mock, write the one-line trap behind every wrong algebra answer. Your list will repeat within a month.
Budget time. With 30 MCQs in 40 minutes, a polynomial question gets about 80 seconds. If you can't see how to start within 30 seconds, mark it and move on.

Conclusion

IPMAT polynomial questions can really help you grab good marks. Learn the six key ideas, solve all questions above with a timer, write down every mistake, and try the ones you got wrong again a week later. IIM Indore has tested this topic every year since 2023, so a few focused weeks of practice can really boost your QA score.

Your next step: See how well you handle polynomials under exam pressure. Take a free SuperGrads IPMAT mock and check your algebra accuracy in the analysis report.

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