10th Class Maths Fully solved Exercise 2.1-2.4 with tips and tricks

Class 10 Maths Chapter 2 – Polynomials (NCERT Detailed Solutions, Concepts, Tricks & Pitfalls)

Prepared by Zayyan Kaseer • Clear, exam-oriented explanations with real-world intuition

Why Polynomials Matter (and How to Ace This Chapter)

Polynomials look simple—just variables and coefficients—but they are the backbone of algebra and a recurring theme in higher mathematics, physics, coding, and data science. Understanding how roots connect to graphs, how coefficients control shape, and how factorisation unlocks hidden structure will make many chapters ahead feel easier. Think of a polynomial as a machine: feed an input x, get an output y. Where the output is zero, that input is special—it’s a zero (root). Graphically, that’s exactly where the curve kisses or crosses the x-axis.

Mindset

Don’t memorise; model. Picture the graph in your head whenever you see an expression. Ask: “What happens as x → +∞? What about −∞? How many times can this curve touch the x-axis?” Visual thinking makes polynomial questions feel natural instead of mechanical.

Jump to Contents Related: Class 9 Maths (Exercise 11–13) More: CBSE 10th Maths Solutions

📑 Table of Contents

• Exercise 2.1 – Geometrical Meaning of Zeroes
• Exercise 2.2 – Relationship Between Zeroes and Coefficients
• Bonus Knowledge
• Masterstroke (Exam-Winning Insight)
• Quick Links (About, Privacy, Contact, T&C)

Exercise 2.1 – Geometrical Meaning of Zeroes

Core Idea in One Line

The zeroes (roots) of a polynomial y=f(x) are precisely the x-coordinates where its graph meets the x-axis (i.e., where y=0).

Analogy

Imagine driving on a hilly road (the graph). Sea level is the x-axis. Wherever the road touches sea level, you’ve hit a zero of the polynomial.

Quick Check

If a curve intersects the x-axis at 2 distinct points, it has 2 real zeroes. If it just touches (tangent) at one point and bounces back, that point is a repeated real zero.

Q1 (Graph Based): Count the Zeroes from y = f(x)

What to do: Look at each provided graph and count how many times it intersects the x-axis.

• 1 intersection → 1 zero
• 2 intersections → 2 zeroes
• 3 intersections → 3 zeroes (possible for cubics)
• No intersection → 0 real zeroes (roots may be complex)

Common Mistake

Students often count peaks or turning points instead of x-axis intersections. Only x-axis crossing/touching points matter.

Deep Insight: Touching vs Crossing

If the curve just touches the x-axis and turns back, the zero at that contact point typically has even multiplicity (like 2). If it crosses through, the zero has odd multiplicity (like 1 or 3). For quadratics:

• Two crossings → two distinct real roots.
• One touch (tangent) → one real repeated root.
• No contact → no real roots (discriminant < 0).

Exam Tip

Mention “curve intersects the x-axis at … points” explicitly. This phrasing earns method marks even if the final count is off by one.

Quick Thought Exercise

1. Sketch a smile-shaped curve (opening up) that never touches the x-axis—what can you say about its zeroes?
2. Now lower the curve until it just kisses the x-axis at one point—what changed about the roots?

This mental model helps you “see” roots before calculating.

Exercise 2.2 – Relationship Between Zeroes and Coefficients

Key Relationships (Quadratic)

For f(x)=ax^2+bx+c with zeroes α and β:

• Sum: α+β = -b/a
• Product: αβ = c/a

Why It Works

Factorising gives a(x-α)(x-β). Expanding and comparing coefficients quickly yields the sum/product formulas—no solving required.

Q1: Find Zeroes and Verify the Relationships

(i) x^2 + 5x + 6

Factorise: Numbers adding to 5 and multiplying to 6 → 2 and 3. So x^2+5x+6=(x+2)(x+3).

Zeroes: x=-2, x=-3

Check: Sum = −5 = −b/a; Product = 6 = c/a ✓

(ii) x^2 - 3x - 10

Factorise: (x − 5)(x + 2). Zeroes 5 and −2. Sum = 3, Product = −10 ✓

(iii) x^2 - 4x + 4

Perfect square: (x − 2)^2 → zeroes 2 and 2 (repeated). Sum = 4, Product = 4 ✓

(iv) 2x^2 + 7x + 3

Factorise: (2x + 1)(x + 3) → zeroes −1/2 and −3. Sum = −7/2, Product = 3/2 ✓

(v) 4s^2 + 4√2 s + 1

Perfect square: (2s + √2)^2 → zeroes both −√2/2. Sum = −√2, Product = 1 ✓

(vi) t^2 - 15

Difference of squares: (t − √15)(t + √15) → zeroes √15 and −√15. Sum = 0, Product = −15 ✓

Common Pitfall

Students sometimes write αβ = a/c by mistake. Always remember: αβ = c/a (constant ÷ leading coefficient).

Q2: Construct Quadratic from Given Sum and Product

Use the template x^2 - (sum)x + (product).

• (i) Sum = −1, Product = 1 → x^2 + x + 1
• (ii) Sum = 1, Product = −6 → x^2 - x - 6
• (iii) Sum = 0, Product = √5 → x^2 - √5
• (iv) Sum = 4, Product = 1 → x^2 - 4x + 1

Exam Tip

Write the template first, then substitute values. This earns steps marks even if arithmetic slips later.

Myth Busting (Be Crystal-Clear)

• Myth: “A quadratic always has 2 real roots.”
  Truth: It always has 2 roots in the complex number system, but they may be complex (non-real). The graph view clarifies this.
• Myth: “If the graph touches the x-axis, that’s two roots.”
  Truth: It’s a repeated root (multiplicity 2) but appears as a single x-value.
• Myth: “Sum and product formulas work only after solving.”
  Truth: They work straight from coefficients—no solving needed.

Mini Quiz (Check Yourself)

1. For 3x^2 - 2x - 5, what are α+β and αβ?
2. If a quadratic just touches the x-axis at x=4, what can you say about its factorisation form?
3. Build the quadratic with sum 7 and product 10.

Answers

1) α+β = -b/a = 2/3, αβ = c/a = -5/3.
2) It has a repeated root: a(x-4)^2 for some a≠0.
3) x^2 - 7x + 10.

Bonus Knowledge — Visual Shortcuts that Save Time

1) Discriminant Snapshot

The discriminant D=b^2-4ac tells you the nature of roots at a glance:

• D>0 → two distinct real roots (two x-axis crossings)
• D=0 → one repeated real root (tangent touch)
• D<0 → no real roots (no contact)

Try This

Before solving, compute D. If it’s not nice, consider factoring tricks or checking if completing the square is quicker.

2) Leading Coefficient = Opening Direction

For ax^2+bx+c, if a>0 the parabola opens up (smile 🙂), if a<0 it opens down (frown 🙃). This instantly hints whether the minimum/maximum is above or below the x-axis.

3) Vertex Coordinates Without Graphing

Vertex at x = -b/(2a), and y = f(-b/(2a)). If the vertex’s y is positive and the curve opens up, there’s no real root; if negative, there are two real roots.

4) Factor or Formula? Decide Fast

• If coefficients are small and product/sum pair is obvious → factorisation.
• If not obvious → quadratic formula or completing the square.
• Always reduce common factors first to simplify arithmetic.

Masterstroke — The “Sum–Product First” Play

When you see ax^2+bx+c and the task involves roots/relations, do this immediately:

1. Write α+β = -b/a, αβ = c/a on the paper.
2. Check if the pair of numbers matching “sum” and “product” is obvious. If yes → factor in 10 seconds.
3. If not obvious → compute D=b^2-4ac and jump to formula. No wasted time.

Why It Wins

This prevents you from getting stuck on non-factorable quadratics and protects your time in competitive exams.

Exercise 2.3 – Division Algorithm for Polynomials

Concept Recap:

The Division Algorithm is one of the most important tools in Polynomials. It says: For any two polynomials p(x) and g(x), where g(x) ≠ 0, there exist unique polynomials q(x) and r(x) such that:

p(x) = g(x) × q(x) + r(x), where degree of r(x) < degree of g(x).

- p(x) → Dividend - g(x) → Divisor - q(x) → Quotient - r(x) → Remainder

Q1: Divide polynomials and verify Division Algorithm:

(i) p(x) = x³ - 3x² + x + 1, g(x) = x² - 2

Step 1: Divide first term x³ ÷ x² = x. Multiply: x(x² - 2) = x³ - 2x. Subtract: (x³ - 3x² + x + 1) - (x³ - 2x) = -3x² + 3x + 1.

Step 2: Divide first term -3x² ÷ x² = -3. Multiply: -3(x² - 2) = -3x² + 6. Subtract: (-3x² + 3x + 1) - (-3x² + 6) = 3x - 5.

So quotient q(x) = x - 3, remainder r(x) = 3x - 5. Verification: x³ - 3x² + x + 1 = (x² - 2)(x - 3) + (3x - 5). ✅

(ii) p(x) = x³ + x² + x + 1, g(x) = x² + 1

Divide: x³ ÷ x² = x. Multiply: x(x² + 1) = x³ + x. Subtract: (x³ + x² + x + 1) - (x³ + x) = x² + 1.

Divide: x² ÷ x² = 1. Multiply: 1(x² + 1) = x² + 1. Subtract: (x² + 1) - (x² + 1) = 0.

So quotient q(x) = x + 1, remainder r(x) = 0. Verification: x³ + x² + x + 1 = (x² + 1)(x + 1) + 0. ✅

🔑 Bonus Knowledge: The Division Algorithm is the backbone of many higher-level algebra concepts, such as finding GCD of polynomials, using Euclid’s Lemma in polynomial form, and simplifying rational expressions. In Class 11 and 12, this idea extends into advanced factorisation and even modular arithmetic.

🎯 Masterstroke for Exams: Always write the verification step after solving – examiners love it and you’ll secure full marks. Even if your quotient/remainder is correct, skipping the verification can cost 1–2 marks in board exams.

Exercise 2.4 – Finding Zeroes of Cubic Polynomials

This exercise is all about extending the idea of zeroes beyond quadratics. Here you’ll learn to find factors of cubic polynomials using the Factor Theorem and then reducing them step by step until you get all three zeroes.

Exercise 2.4 – Finding Zeroes of Cubic Polynomials

Concept Recap:

A cubic polynomial has the general form ax³ + bx² + cx + d, where a ≠ 0. Its zeroes are values of x for which the polynomial equals 0.

To find zeroes, we generally follow these steps:

1. Check for obvious rational roots using the Rational Root Theorem.
2. Use Factor Theorem to factor out one linear factor.
3. Reduce cubic to quadratic by dividing polynomial by linear factor.
4. Solve remaining quadratic to get other zeroes.

Q1: Find zeroes of x³ - 6x² + 11x - 6

Step 1: Check for rational roots: possible factors of 6 (constant term) over factors of 1 (coefficient of x³) → ±1, ±2, ±3, ±6.

Step 2: Test x = 1 → f(1) = 1 - 6 + 11 - 6 = 0 ✅ So x - 1 is a factor.

Step 3: Divide cubic by (x - 1) → Using synthetic division or long division:

x³ - 6x² + 11x - 6 ÷ (x - 1) = x² - 5x + 6

Step 4: Factor quadratic: x² - 5x + 6 = (x - 2)(x - 3)

Zeroes: x = 1, 2, 3 ✅ Verified

Q2: Find zeroes of 2x³ + x² - 8x - 4

Step 1: Rational Root Theorem → factors of -4 over factors of 2: ±1, ±2, ±4, ±1/2, ±2

Step 2: Test x = -1 → f(-1) = -2 +1 +8 -4 = 3 ❌ Not zero

Step 3: Test x = 1 → f(1) = 2 +1 -8 -4 = -9 ❌

Step 4: Test x = -2 → f(-2) = -16 +4 +16 -4 = 0 ✅ So x + 2 is a factor

Step 5: Divide cubic by (x + 2) → quotient = 2x² - 3x - 2

Step 6: Factor quadratic → 2x² - 3x - 2 = (2x + 1)(x - 2)

Zeroes: x = -2, -1/2, 2 ✅ Verified

🔑 Bonus Knowledge: Using synthetic division speeds up finding zeroes tremendously. Memorize common patterns for cubics like sum of zeroes = -b/a, product of zeroes = -d/a, which helps in double-checking your solutions.

🎯 Masterstroke Tip: Always start testing small integers (±1,2,3…) first. Most NCERT problems are designed with small integer zeroes. This avoids unnecessary complexity and saves time during exams.

Q3: Solve 3x³ - x² - 11x + 3

Step 1: Rational roots: factors of 3 / factors of 3 = ±1, ±3, ±1/3

Step 2: Test x = 3 → f(3) = 81 - 9 - 33 + 3 = 42 ❌

Test x = 1 → f(1) = 3 -1 -11 +3 = -6 ❌

Test x = -1 → f(-1) = -3 -1 +11 +3 = 10 ❌

Test x = 1/3 → f(1/3) = 3*(1/27) -1/9 -11/3 +3 = 1/9 -1/9 -11/3 +3 = -11/3 +3 = -2/3 ❌

Test x = -1/3 → f(-1/3) = -1/9 -1/9 +11/3 +3 = 0 ✅ So x + 1/3 is a factor

Step 3: Divide cubic by (3x + 1) → quotient = x² - 4x + 3

Step 4: Factor quadratic → (x -1)(x -3)

Zeroes: x = -1/3, 1, 3 ✅ Verified

Interactive Thought Exercise:

Visualize a cubic graph. Note how zeroes represent x-intercepts. The nature of turning points also tells you whether roots are repeated or distinct. Try sketching each example above to internalize the concept visually.

🔑 Advanced Bonus: Remember Vieta’s formulas for cubic: - Sum of zeroes: α + β + γ = -b/a - Sum of products of two zeroes: αβ + βγ + γα = c/a - Product of zeroes: αβγ = -d/a These formulas help verify your answers quickly.

🎯 Exam Masterstroke: Cross-check sum/product relations after finding zeroes. This extra step ensures no calculation mistakes and can save marks in high-stakes exams.

Forward-Looking Insight:

Mastery of cubic polynomials sets a strong foundation for advanced algebra in Class 11, 12, and competitive exams. Recognizing patterns in zeroes, factorization shortcuts, and synthetic division are essential skills for higher mathematics and problem-solving efficiency.

💡 Motivational Message from Zayyan Kaseer: Remember, mathematics is like a puzzle – each zero, each factor is a piece. With patience and practice, you can see the full picture clearly. Keep exploring, keep solving, and trust your reasoning skills – every step you take strengthens your mastery.

Q1: What are polynomials?

Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents.

Q2: How do we find the zeroes of a quadratic polynomial?

Zeroes can be found by factoring the polynomial, using the quadratic formula, or completing the square. These are the x-values where the polynomial equals zero.

Q3: Can a polynomial have complex zeroes?

Yes, polynomials with real coefficients can have complex zeroes, which always occur in conjugate pairs.

Q4: What is the relationship between zeroes and coefficients?

For a quadratic polynomial ax² + bx + c, sum of zeroes α + β = -b/a, and product αβ = c/a.

Q5: Why is understanding zeroes important?

Zeroes help in sketching graphs, solving equations, and understanding polynomial behavior in real-world problems.

Disclaimer: All information and educational content is for general knowledge only and not a substitute for professional advice. Use responsibly.
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