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Polynomials

NCERT Class 10 · Mathematics Based on NCERT Class 10 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter builds on the concept of polynomials introduced in Class IX, focusing on linear, quadratic, and cubic polynomials. The central idea is understanding the relationship between the zeroes of a polynomial and its coefficients, along with the geometrical meaning of zeroes. A student must walk away knowing how to find zeroes, verify relationships, and apply the sum and product of zeroes to form quadratic polynomials.
Key Concepts (12)
Polynomial
An expression of the form p(x) where x is a variable, and the highest power of x is called the degree of the polynomial.
e.g. 5x³ – 4x² + x – 2 is a polynomial in x of degree 3.
Degree of a polynomial
The highest power of the variable in the polynomial.
e.g. In 7u⁶ – 4u³ + u² + 2u – 8, the degree is 6.
Linear polynomial
A polynomial of degree 1.
e.g. 2x – 3, 3x + 5, 2y + √2 are linear polynomials.
Quadratic polynomial
A polynomial of degree 2, generally of the form ax² + bx + c, where a, b, c are real numbers and a ≠ 0.
e.g. x² – 3x – 4 is a quadratic polynomial.
Cubic polynomial
A polynomial of degree 3, generally of the form ax³ + bx² + cx + d, where a, b, c, d are real numbers and a ≠ 0.
e.g. 2x³ – 5x² – 14x + 8 is a cubic polynomial.
Value of a polynomial at x = k
The value obtained by replacing x by k in p(x), denoted by p(k).
e.g. For p(x) = x² – 3x – 4, p(2) = 2² – 3×2 – 4 = –6.
Zero of a polynomial
A real number k is said to be a zero of a polynomial p(x) if p(k) = 0.
e.g. For p(x) = x² – 3x – 4, p(–1) = 0 and p(4) = 0, so –1 and 4 are zeroes.
Zero of a linear polynomial
For p(x) = ax + b, the zero is given by k = –b/a.
e.g. For p(x) = 2x + 3, zero is –3/2.
Geometrical meaning of zeroes
The zeroes of a polynomial are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
e.g. For y = x² – 3x – 4, the graph cuts x-axis at x = –1 and x = 4.
Parabola
The shape of the graph of a quadratic polynomial y = ax² + bx + c; opens upwards if a > 0 and opens downwards if a < 0.
e.g. y = x² – 3x – 4 is a parabola opening upwards.
Relationship between zeroes and coefficients of a quadratic polynomial
If α and β are zeroes of ax² + bx + c, then α + β = –b/a and αβ = c/a.
e.g. For 2x² – 8x + 6, zeroes are 1 and 3; sum = 4 = –(–8)/2, product = 3 = 6/2.
Relationship between zeroes and coefficients of a cubic polynomial
If α, β, γ are zeroes of ax³ + bx² + cx + d, then α + β + γ = –b/a, αβ + βγ + γα = c/a, and αβγ = –d/a.
e.g. For 3x³ – 5x² – 11x – 3, zeroes are 3, –1, –1/3; sum = 5/3 = –(–5)/3.
Formulas & Equations (6)
Zero of linear polynomial ax + b: k = –b/a
Reads as: k equals negative b over a
Variables: a = coefficient of x (a ≠ 0), b = constant term
Use when: Finding the zero of a linear polynomial in one variable.
⚠ Students often write k = b/a, forgetting the negative sign.
Sum of zeroes of quadratic polynomial: α + β = –b/a
Reads as: alpha plus beta equals negative b over a
Variables: α, β = zeroes of polynomial; a = coefficient of x²; b = coefficient of x
Use when: Verifying relationship between zeroes and coefficients, or finding sum of zeroes given coefficients.
⚠ Writing α + β = b/a without the negative sign.
Product of zeroes of quadratic polynomial: αβ = c/a
Reads as: alpha beta equals c over a
Variables: α, β = zeroes; a = coefficient of x²; c = constant term
Use when: Verifying relationship or finding product of zeroes.
⚠ Confusing c with the coefficient of x (b).
Sum of zeroes of cubic polynomial: α + β + γ = –b/a
Reads as: alpha plus beta plus gamma equals negative b over a
Variables: α, β, γ = zeroes; a = coefficient of x³; b = coefficient of x²
Use when: Verifying relationship for cubic polynomials.
⚠ Forgetting the negative sign or using wrong coefficient.
Sum of products of zeroes taken two at a time (cubic): αβ + βγ + γα = c/a
Reads as: alpha beta plus beta gamma plus gamma alpha equals c over a
Variables: α, β, γ = zeroes; a = coefficient of x³; c = coefficient of x
Use when: Verifying the second relationship for cubic polynomials.
⚠ Mixing up c with constant term d.
Product of zeroes of cubic polynomial: αβγ = –d/a
Reads as: alpha beta gamma equals negative d over a
Variables: α, β, γ = zeroes; a = coefficient of x³; d = constant term
Use when: Verifying the third relationship for cubic polynomials.
⚠ Writing αβγ = d/a without the negative sign.
How It Works
Finding zeroes of a quadratic polynomial by splitting the middle term
  1. Write the quadratic polynomial in the form ax² + bx + c.
  2. Find two numbers whose sum is b and product is a × c.
  3. Split the middle term bx into two terms using these numbers.
  4. Factor by grouping to get the product of two linear factors.
  5. Set each factor equal to zero and solve for x to get the zeroes.
💡 Students often forget to check that the sum of the two numbers equals b, not –b.
Verifying relationship between zeroes and coefficients of a quadratic polynomial
  1. Find the zeroes α and β of the polynomial.
  2. Compute α + β and compare with –b/a.
  3. Compute αβ and compare with c/a.
  4. If both match, the relationship is verified.
💡 Always use the standard form ax² + bx + c; do not forget the negative sign in the sum formula.
Finding a quadratic polynomial given sum and product of zeroes
  1. Let the zeroes be α and β.
  2. Given α + β = S and αβ = P.
  3. The polynomial is of the form k[x² – (α + β)x + αβ].
  4. For k = 1, the polynomial is x² – Sx + P.
  5. Any real k gives a family of polynomials satisfying the conditions.
💡 Students often write x² + Sx + P instead of x² – Sx + P; the sign of the linear term is negative.
Verifying relationship between zeroes and coefficients of a cubic polynomial
  1. Find the three zeroes α, β, γ of the cubic polynomial.
  2. Compute α + β + γ and compare with –b/a.
  3. Compute αβ + βγ + γα and compare with c/a.
  4. Compute αβγ and compare with –d/a.
  5. If all match, the relationship is verified.
💡 The product formula has a negative sign: αβγ = –d/a, not d/a.
Exam Traps — Don't Make These Mistakes
⚠ Forgetting the negative sign in sum of zeroes
✗ α + β = b/a
→ The correct formula is α + β = –b/a; the negative sign comes from comparing coefficients of the factorised form.
✓ α + β = –b/a
⚠ Confusing constant term with coefficient of x in product formula
✗ αβ = b/a
→ The product of zeroes is c/a, where c is the constant term, not the coefficient of x.
✓ αβ = c/a
⚠ Writing the quadratic polynomial as x² + Sx + P instead of x² – Sx + P
✗ x² + (α + β)x + αβ
→ From the factorised form (x – α)(x – β) = x² – (α + β)x + αβ, the linear term has a negative sign.
✓ x² – (α + β)x + αβ
⚠ Mistaking the zero of a linear polynomial as b/a
✗ Zero = b/a
→ Setting ax + b = 0 gives x = –b/a, not b/a.
✓ Zero = –b/a
⚠ Forgetting the negative sign in product of zeroes of a cubic polynomial
✗ αβγ = d/a
→ The correct relationship is αβγ = –d/a, derived from comparing constant terms.
✓ αβγ = –d/a
⚠ Assuming all quadratic polynomials have real zeroes
✗ Every quadratic polynomial has two zeroes.
→ A quadratic polynomial can have two distinct zeroes, two equal zeroes, or no real zeroes (if the graph does not intersect the x-axis).
✓ A quadratic polynomial can have at most two zeroes; it may have two distinct, two equal, or no real zeroes.
Compare & Contrast
Degree
Linear polynomial
Quadratic polynomial
Linear polynomial has degree 1, while quadratic polynomial has degree 2.
Formula
Sum of zeroes of quadratic polynomial
Sum of zeroes of cubic polynomial
For quadratic, α + β = –b/a; for cubic, α + β + γ = –b/a. Both involve the negative of the coefficient of the second highest degree term divided by the leading coefficient.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Short Answer (2-3 marks), 1 Long Answer (4-5 marks)
Most asked: Finding zeroes of quadratic polynomials and verifying relationship between zeroes and coefficients · Finding a quadratic polynomial given sum and product of zeroes · Geometrical meaning of zeroes (number of zeroes from graph)
Don't skip: Verify relationship between zeroes and coefficients of a given quadratic polynomial (5 marks); Find a quadratic polynomial when sum and product of zeroes are given (3 marks)
One-Line Revision (read aloud before the exam)
  1. A polynomial of degree 1 is linear, degree 2 is quadratic, and degree 3 is cubic.
  2. A real number k is a zero of polynomial p(x) if p(k) = 0.
  3. Zero of linear polynomial ax + b is –b/a.
  4. The graph of a linear polynomial is a straight line intersecting the x-axis at exactly one point.
  5. The graph of a quadratic polynomial is a parabola; it can intersect the x-axis at 0, 1, or 2 points.
  6. A quadratic polynomial can have at most two zeroes.
  7. A cubic polynomial can have at most three zeroes.
  8. For quadratic polynomial ax² + bx + c, sum of zeroes = –b/a and product = c/a.
  9. For cubic polynomial ax³ + bx² + cx + d, sum of zeroes = –b/a, sum of products taken two at a time = c/a, and product = –d/a.
  10. To find a quadratic polynomial given sum S and product P of zeroes, use x² – Sx + P.
  11. The zeroes of a polynomial are the x-coordinates where its graph cuts the x-axis.
  12. Expressions like 1/(x–1) and √(x+2) are not polynomials.
  13. The constant term of a quadratic polynomial divided by the coefficient of x² gives the product of its zeroes.
  14. Splitting the middle term is a method to factorise quadratic polynomials and find zeroes.

MCQs — 10 Questions with Answers

Q1. A student observes that the graph of y = 2x + 3 crosses the x-axis at x = −3/2. Which statement best explains why this x-value is called a zero of the polynomial 2x + 3?

  • A. Because substituting x = −3/2 into 2x + 3 gives 0 ✓
  • B. Because the graph touches the x-axis at that point
  • C. Because the slope of the line is 2
  • D. Because −3/2 is a negative number

Answer: A — A zero is defined as a value of x where p(x) = 0, not merely where the graph intersects the x-axis; option B describes the geometric consequence but not the algebraic definition, which is what 'zero' means.

Q2. Assertion (A): A quadratic polynomial ax² + bx + c with a > 0 can have no real zeroes. Reason (R): When the parabola y = ax² + bx + c opens upward and does not intersect the x-axis, the polynomial has no real zeroes. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both statements are true; R directly explains why A is true because the geometric fact (no x-intercept) is the reason the algebraic fact (no real zeroes) holds.

Q3. A linear polynomial p(x) = ax + b has exactly one zero at x = k. Which of the following must be true?

  • A. The graph of y = ax + b is a horizontal line
  • B. The point (k, 0) lies on the graph of y = ax + b ✓
  • C. The value of b is always negative
  • D. The slope a must be greater than 1

Answer: B — If k is a zero, then p(k) = 0, which means the point (k, 0) satisfies the equation y = ax + b; a horizontal line would have infinitely many zeroes or none, and b and a have no fixed sign restrictions.

Q4. Consider two quadratic polynomials: p(x) = x² − 5x + 6 and q(x) = −x² + 5x − 6. A student claims that p(x) and q(x) have the same zeroes because one is the negative of the other. Is this claim always true, sometimes true, or never true?

  • A. Always true, because if p(k) = 0 then −p(k) = 0 for any polynomial p
  • B. Sometimes true, only when the polynomial is monic
  • C. Never true, because the zeroes of q(x) are the opposites of the zeroes of p(x)
  • D. Always true, because negating a polynomial does not change where it crosses the x-axis ✓

Answer: D — If p(k) = 0, then −p(k) = 0 as well, so q(k) = −p(k) = 0; the zeroes are identical, not opposite (verify: p(x) = 0 at x = 2, 3; q(x) = 0 also at x = 2, 3).

Q5. Assertion (A): A cubic polynomial ax³ + bx² + cx + d always has at least one real zero. Reason (R): A cubic polynomial has degree 3, which is odd, so its graph must cross the x-axis at least once. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both are true; the odd degree ensures the polynomial takes both positive and negative values, so by continuity it must cross the x-axis, which is exactly why it has at least one real zero.

Q6. A quadratic polynomial has zeroes at x = 2 and x = 5. Which statement correctly relates the zeroes to the polynomial's properties?

  • A. The polynomial must be of the form a(x − 2)(x − 5) for some non-zero constant a ✓
  • B. The polynomial must be x² − 7x + 10
  • C. The polynomial's graph must open upward
  • D. The polynomial's y-intercept must be positive

Answer: A — Any quadratic with zeroes 2 and 5 can be written as a(x − 2)(x − 5), where a ≠ 0 is arbitrary; option B fixes a = 1, option C requires a > 0, and option D requires 10a > 0, all of which are unnecessary restrictions.

Q7. Assertion (A): If p(x) is a polynomial and p(k) = 0, then (x − k) is a factor of p(x). Reason (R): The zero of a linear polynomial ax + b is always x = −b/a. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A
  • B. Both A and R are true but R is not the correct explanation of A ✓
  • C. A is true but R is false
  • D. A is false but R is true

Answer: B — Both are true statements, but R is about linear polynomials only and does not explain A, which is a general statement about any polynomial; A is known as the Factor Theorem.

Q8. A student graphs y = x² − 4 and observes it crosses the x-axis at x = −2 and x = 2. Based on this observation, which algebraic fact does the student verify?

  • A. (−2)² − 4 = 0 and (2)² − 4 = 0 ✓
  • B. The polynomial x² − 4 can be factored as (x − 2)(x + 2)
  • C. The coefficients of x² − 4 sum to zero
  • D. The graph of x² − 4 is symmetric about the y-axis

Answer: A — The x-intercepts directly verify that p(−2) = 0 and p(2) = 0, which is the definition of zeroes; while B, C, and D are related properties, they are not directly verified by observing where the graph crosses the x-axis.

Q9. Assertion (A): Every polynomial of degree n has at most n zeroes. Reason (R): A polynomial can be written as a product of linear factors corresponding to its zeroes. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A
  • B. Both A and R are true but R is not the correct explanation of A ✓
  • C. A is true but R is false
  • D. A is false but R is true

Answer: B — Both statements are true, but R describes factorization while A is a statement about the maximum count of zeroes; R is related but does not fully explain why the maximum is exactly n (the explanation requires the Fundamental Theorem of Algebra).

Q10. A polynomial p(x) has the property that p(0) = 5. Based solely on this information, which conclusion is valid?

  • A. The polynomial has a zero at x = 5
  • B. The constant term of the polynomial is 5 ✓
  • C. The polynomial is of degree 1
  • D. The polynomial cannot have any real zeroes

Answer: B — p(0) gives the y-intercept, which equals the constant term; p(0) = 5 tells us nothing about whether 5 is a zero (it is not, since p(5) is unknown), the degree, or the existence of other zeroes.

Flashcards

What is the definition of a zero of polynomial p(x)?

A real number k is a zero of p(x) if p(k) = 0.

What is the general form of a quadratic polynomial?

ax² + bx + c, where a, b, c are real numbers and a ≠ 0.

How many zeroes can a quadratic polynomial have at most?

A quadratic polynomial can have at most 2 zeroes.

What does a zero of a polynomial represent on its graph?

A zero of p(x) is the x-coordinate of the point where the graph intersects the x-axis.

If p(x) = 2x + 3, what is its zero?

The zero is -3/2, found by solving 2x + 3 = 0.

What is the shape of the graph of a quadratic polynomial ax² + bx + c when a > 0?

The parabola opens upwards.

What is the difference between a cubic polynomial and a quadratic polynomial?

A cubic polynomial has degree 3, while a quadratic has degree 2; a cubic can have up to 3 zeroes.

When does a quadratic polynomial have exactly one zero?

When the parabola touches the x-axis at exactly one point (two coincident zeroes).

For linear polynomial ax + b (a ≠ 0), what is the zero?

The zero is -b/a, which equals -(Constant term)/(Coefficient of x).

Can a quadratic polynomial have no real zeroes? If yes, what does this mean graphically?

Yes; the parabola is completely above or below the x-axis and never intersects it.

Important Board Questions

Define the zero of a polynomial. Is –2 a zero of the polynomial p(x) = 3x² + 5x – 2? Justify your answer. [2 marks]

Zero is defined as k where p(k) = 0. Calculate p(–2) = 3(4) + 5(–2) – 2 and check if it equals zero.

Find the zeroes of the quadratic polynomial x² – 2x – 8 and verify that they are the x-coordinates of the points where the graph intersects the x-axis. [3 marks]

Factor x² – 2x – 8 = (x – 4)(x + 2) to find zeroes 4 and –2. Verify by substituting: p(4) = 0 and p(–2) = 0; these x-values are where the parabola touches the x-axis.

Explain with the help of a graph why a quadratic polynomial can have 0, 1, or 2 zeroes. How does the position of the parabola relative to the x-axis determine the number of zeroes? Also, state how many zeroes a cubic polynomial can have at most. [5 marks]

Describe three cases: (1) parabola cuts x-axis at two points → 2 distinct zeroes; (2) parabola touches x-axis at one point → 1 zero (repeated); (3) parabola lies completely above/below x-axis → no zeroes. Explain that 'a' (coefficient of x²) determines if parabola opens up/down. A cubic polynomial of degree 3 has at most 3 zeroes by the fundamental theorem.

Assertion-Reason Questions for Polynomials

Q1. Assertion (A):

A polynomial of degree 2 is called a quadratic polynomial and has the general form ax² + bx + c where a ≠ 0.

Reason (R):

A quadratic polynomial must have its highest power equal to 2, and the coefficient of x² cannot be zero.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The assertion correctly defines a quadratic polynomial with the standard form. The reason is the correct explanation because it explains why 'a ≠ 0' is required (to ensure the highest power is actually 2, not less). This is a fundamental definition from the chapter.

Q2. Assertion (A):

The zeroes of a quadratic polynomial ax² + bx + c are the x-coordinates of the points where the parabola y = ax² + bx + c intersects the x-axis.

Reason (R):

A zero of a polynomial p(x) is defined as any value k where p(k) = 0, and on the graph, this corresponds to y = 0, which is the x-axis.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains the assertion. When p(x) = 0, the point (x, 0) lies on both the curve and the x-axis. The zeroes are precisely the x-coordinates of intersection points with the x-axis, which is what the reason establishes.

Q3. Assertion (A):

A quadratic polynomial can have at most 3 distinct zeroes.

Reason (R):

The number of zeroes of a polynomial is determined by its degree, and a polynomial of degree n has at most n zeroes.

Answer: D — A is false but R is true — The assertion is false. A quadratic polynomial has degree 2, so it can have at most 2 zeroes, not 3. While the reason statement is generally true (a polynomial of degree n has at most n zeroes), it actually contradicts the assertion rather than supporting it. Since the assertion is false, the answer is D.

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Case-Based Questions (CBQ) for Polynomials

Case Study:

A manufacturing company produces cylindrical containers where the volume V of each container is given by the polynomial function V(r) = πr² × h, where r is the radius and h is the height (both in cm). For a specific production line, the height is fixed at 10 cm. The volume function becomes V(r) = 10πr². A quality control engineer needs to find the radius values that would result in zero volume (which occurs only when radius is zero, indicating a degenerate case). The engineer is studying various radius values: r = 0, r = 2, r = 3, and r = 5 cm, and calculating corresponding volumes to understand the relationship between radius changes and volume output.

Q1. What is the degree of the polynomial function V(r) = 10πr² with respect to r?

  • A. 1
  • B. 2 ✓
  • C. 3
  • D. 0

Answer: B

Q2. What is V(0) for the volume function V(r) = 10πr²?

  • A. 10π
  • B. 0 ✓
  • C. 20π
  • D. 100π

Answer: B

Q3. Calculate V(2) for V(r) = 10πr².

  • A. 40π ✓
  • B. 20π
  • C. 60π
  • D. 80π

Answer: A

Q4. What is the zero of the polynomial V(r) = 10πr²?

  • A. r = 0 only ✓
  • B. r = 0 and r = 10
  • C. r = 1
  • D. No real zero exists

Answer: A

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Previous Year Questions (PYQ) for Polynomials

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 31/4/1 · 1 mark

The graph of polynomial p(x) = k is shown here. Number of zeroes of polynomial p(x) is:

  • 0
  • 1
  • 2
  • infinitely many

CBSE 2026 · Set 31/4/1 · 2 marks

If α, β are the zeroes of polynomial p(x) = 6x² – 5x – 3, then find the value of 1/α + 1/β.

CBSE 2026 · Set 31/4/2 · 2 marks

If α and β are the zeroes of polynomial p(x) = – 9x² – 6x + 1. Find the value of α² + β².

CBSE 2026 · Set 31/4/3 · 2 marks

Form a quadratic polynomial whose sum and product of the zeroes are 1/2 and –1/9 respectively. Hence, find the zeroes of the polynomial.

CBSE 2026 · Set 31/5/1 · 1 mark

The graph of a polynomial p(x) is shown here. The number of zeroes of the polynomial p(x) is

  • 5
  • 1
  • 0
  • 4

CBSE 2026 · Set 31/5/1 · 1 mark

The value of k for which sum of the zeroes of the polynomial p(x) = 3x² – kx + 6 is 2, is

  • 2
  • –6
  • –2
  • 6

CBSE 2026 · Set 31/5/1 · 3 marks

If α and β are zeroes of the polynomial p(x) = 5x² – 7x – 3, then form a quadratic polynomial whose zeroes are α² and β².

CBSE 2026 · Set 31/5/2 · 1 mark

If the zeroes of the polynomial p(x) = 2x² – 7x + 6 are α and β, then the value of 1/α + 1/β is

  • 7/2
  • 6/7
  • –7/6
  • 7/6

CBSE 2026 · Set 31/5/2 · 3 marks

(a) Find the zeroes of the polynomial p(x) = 4x² – 8x + 3 and verify the relationship between its zeroes and co-efficients. OR (b) Form a polynomial whose zeroes are 2α and 2β, where α and β are zeroes of the polynomial p(x) = x² – 3√2x + 4.

CBSE 2026 · Set 31/5/3 · 3 marks

If α, β are zeroes of the polynomial p(x) = 5x² – 7x – 3, then form a quadratic polynomial whose zeroes are α/2 and β/2.

CBSE 2026 · Set 31/1164/30 · 1 mark

At which of the following points, the quadratic polynomial p(x) = – 3x + 18x2 – 1 intersects the positive x-axis ?

  • (1/6, 0)
  • (1/3, 0)
  • (1/6, 0)
  • (1/3, 0)

CBSE 2026 · Set 31/1164/30 · 1 mark

The graph of a quadratic polynomial f(x) passes through (5,0), (0, –1) and (–2, 0). The two factors of the polynomial are

  • ( x + 2), (x – 5)
  • ( x + 5), (x – 2)
  • ( x + 1), (x – 5)
  • ( x – 1), (x + 2)

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Class 10 Maths Formula Sheet

Every formula from all 14 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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Class 10 Mathematics Previous Year Questions, Chapter-wise

All 1107 CBSE board questions from 2022–2026, sorted by chapter and question type. Free PDF download.

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