NCERT Exemplar Class 10 Maths Polynomials Exercise 2.4 Question 4 Step-by-Step Solution (with Video)
If you are preparing for the CBSE Class 10 Maths board exam, this question is a must-solve. It combines three of the highest-weightage concepts from Chapter 2 the factor theorem, polynomial division, and zeroes of a polynomial into a single problem.
The Question (Exercise 2.4, Q4)
Find the value of k so that x² + 2x + k is a factor of 2x⁴ + x³ − 14x² + 5x + 6. Also find all the zeroes of the two polynomials.
QUICK ANSWER
Value of k = −3
Zeroes of x² + 2x − 3: x = 1, −3
Zeroes of 2x⁴ + x³ − 14x² + 5x + 6: x = 1, −3, −½, 2
Now let’s understand how we arrived at this answer — because in the board exam, you’ll be marked on the steps, not just the final value.
Concept Recap — What You Need to Know Before Solving
Before jumping into the solution, make sure you’re solid on these three Class 10 Polynomials concepts:
1. The Factor Theorem
If a polynomial g(x) is a factor of another polynomial p(x), then when you divide p(x) by g(x), the remainder is zero. This is the heart of today’s question.
2. Polynomial Long Division
Just like you divide numbers, you can divide polynomials. The result follows the Division Algorithm:
When the divisor is a factor of the dividend, the remainder is zero — and that gives us an equation we can solve for k.
3. Zeroes of a Polynomial
A zero of a polynomial p(x) is a value of x for which p(x) = 0. Once you factorise a polynomial into smaller factors, each factor gives you its own zeroes.
Step-by-Step Solution
Step 1 — Set up the polynomial long division.
We need to divide p(x) = 2x⁴ + x³ − 14x² + 5x + 6 by g(x) = x² + 2x + k. Since g(x) is a factor of p(x), the final remainder must be zero.
Step 2 — Perform the long division carefully.
After dividing, you’ll get a quotient of the form 2x² + ax + b (where the values depend on k), and a remainder that contains the variable k. The key trick: collect all terms involving k on one side.
Step 3 — Set the remainder equal to zero.
Because g(x) divides p(x) exactly, both the coefficient of x in the remainder and the constant term in the remainder must individually equal zero. This gives us two equations in k.
Step 4 — Solve for k.
Solving the system of equations gives us a single consistent value: k = −3. So the factor becomes x² + 2x − 3.
Step 5 — Factorise x² + 2x − 3 to find the first two zeroes.
Split the middle term: x² + 2x − 3 = x² + 3x − x − 3 = x(x + 3) − 1(x + 3) = (x + 3)(x − 1).
So the zeroes of x² + 2x − 3 are x = 1 and x = −3.
Step 6 — Find the remaining zeroes of the bigger polynomial.
Substituting k = −3 back, the quotient of the division becomes 2x² − 3x − 2. Factorising: 2x² − 3x − 2 = 2x² − 4x + x − 2 = 2x(x − 2) + 1(x − 2) = (2x + 1)(x − 2).
So the remaining zeroes are x = −½ and x = 2.
Step 7 — State all zeroes of 2x⁴ + x³ − 14x² + 5x + 6.
All four zeroes are: 1, −3, −½, and 2.
Summary Table — Both Polynomials at a Glance
🎯 Key Takeaways for the Board Exam
- Factor theorem rule: If g(x) is a factor of p(x), the remainder of p(x) ÷ g(x) must be zero.
- The “find k” trick: Set every coefficient of the remainder equal to zero — that’s where your equation for k comes from.
- Always verify: Substitute k back into the divisor and check that both polynomials factorise cleanly.
- Don’t skip steps: The board exam awards step-marking, so write the division, the remainder equations, and the factorisations clearly.
- Practice this exact type: NCERT Exemplar Polynomials Ex 2.4 has multiple variations of this pattern.
Common Mistakes Students Make in This Question
Even toppers slip up on this one. Watch out for these:
- Sign errors during long division. When you subtract polynomials, students often forget to flip the sign of every term — losing 2–3 marks unnecessarily.
- Forgetting that the remainder must be identically zero. It’s not enough for just the constant to be zero — every coefficient must independently equal zero.
- Wrong factorisation of 2x² − 3x − 2. Many students try 2x² − x − 2 instead. Always re-check by multiplying back.
- Not stating all zeroes clearly at the end. The question asks for zeroes of both polynomials — write them separately and label them.
- Skipping verification. Plug at least one zero back into the original polynomial to confirm p(x) = 0 before moving on.
Why This Question Matters for CBSE Class 10 Board Exam 2026
The NCERT Exemplar is not your regular textbook — it’s the bridge between basic NCERT problems and the higher-order thinking (HOTS) questions that appear in the CBSE Class 10 Maths board exam. Every year, the board picks at least one Polynomials question that uses this exact pattern:
- A polynomial of degree 4 (biquadratic) with a quadratic factor
- An unknown coefficient to be found using the factor theorem
- A request to find all the zeroes
Recent CBSE sample papers and board papers from the last 5 years have repeatedly featured questions modelled on this exemplar problem. Solve this, and you’ve covered a 3-to-5-mark question that has a very high probability of appearing on your board paper.
📘 Want the Full Polynomials Mastery?
Get the complete Class 10 Polynomials chapter every NCERT and Exemplar question solved step-by-step, plus a chapter-wise test series.
Frequently Asked Questions
What is the value of k if x² + 2x + k is a factor of 2x⁴ + x³ − 14x² + 5x + 6?
The value of k is −3. We find this by dividing the larger polynomial by x² + 2x + k. Since the smaller polynomial is a factor, the remainder must be zero. Setting each coefficient of the remainder to zero gives k = −3.
What are all the zeroes of 2x⁴ + x³ − 14x² + 5x + 6?
The four zeroes are 1, −3, −½, and 2. The factorised form is (x + 3)(x − 1)(2x + 1)(x − 2).
How do I find k when one polynomial is a factor of another in Class 10?
Use polynomial long division. Divide the bigger polynomial by the smaller one. Since the smaller one is a factor, the remainder is zero. Equate each coefficient of the remainder to zero — those equations give you the value of k.
Is NCERT Exemplar Exercise 2.4 Q4 important for the CBSE Class 10 board exam?
Yes — extremely important. This question type has appeared in CBSE board exams and sample papers repeatedly. It tests factor theorem, polynomial long division, and finding zeroes — three of the highest-weightage concepts in Chapter 2 Polynomials.
What is the factor theorem in simple words?
If you put a number into a polynomial and the polynomial equals zero, then (x − that number) is a factor of the polynomial. For example, if p(2) = 0, then (x − 2) is a factor of p(x).
How do I verify my answer for this question?
Substitute each zero (1, −3, −½, 2) into the original polynomial 2x⁴ + x³ − 14x² + 5x + 6. If the polynomial equals zero for each substitution, your answer is correct.
Where can I find video solutions for the full NCERT Exemplar Class 10 Polynomials chapter?
The complete NCERT Exemplar Polynomials playlist by Aman Sir is available on the Maths Vidya Institute YouTube channel. Watch the full Class 10 Polynomials playlist here.
Related Lessons You Should Watch Next
- Class 10 Polynomials — Full Chapter Video Course
- NCERT Exemplar Class 10 Polynomials — Complete Playlist
- Class 10 Chapter-wise Test Series (Including Polynomials)
- Class 10 NCERT Solutions — All Chapters
Need 1-on-1 Help from Aman Sir?
Book a free live demo class and see Aman Sir’s teaching style the same one that helped 100+ students score 95+ in CBSE Class 10 Maths boards.
Founder, Maths Vidya Institute · B.Tech Engineer · 15+ years teaching CBSE Class 8–12 Maths. Read full bio →
Last updated: May 2026 · Subject: CBSE Class 10 Maths · Chapter: Polynomials · Source: NCERT Exemplar Mathematics Class 10
