Polynomials Class 10 Maths Worksheet with Answers Chapter 2

Now we are in the 2nd chapter of the NCERT class 10 mathematics book. For the Polynomials chapter, we advise you to give it some time and solve the questions in the textbook properly. Attempt the worksheet only after completing the chapter. This worksheet has four sections: Basic, Standard, Advance, and HOTS.

The topics covered in the chapter –

• Polynomial definitions basics
• Degree and terms
• Zeroes and graphs
• Factor and remainder
• Zeroes–coefficients relation

Class 10 Maths Worksheet – Chapter 2: Polynomials

Basic

1. Write the meaning of degree of a polynomial (1–2 lines).
2. Identify the degree and type (linear/quadratic/cubic):
   • p(x) = 4x + 2
   • q(x) = 5x² − 4x + 1
   • r(x) = 3x³ − 2x² + x − 1
3. For p(x) = x² − 3x − 4, find:
   • p(2)
   • p(0)
   • p(−1)
4. Define zero of a polynomial in one sentence using p(k) = 0.
5. Find the zero of the linear polynomial 2x + 3.
6. If the graph of y = p(x) intersects the x-axis at exactly 3 points, then what is the maximum possible degree of p(x)?
7. Complete the table for p(x) = x² − 3x − 4:

   x	−2	−1	0	1	2
   p(x)	_____	_____	_____	_____	_____

Standard

1. Find the zeroes of x² + 7x + 10, and verify:
   (sum of zeroes) = −(coefficient of x)/(coefficient of x²)
   (product of zeroes) = (constant term)/(coefficient of x²)
2. Find the zeroes of x² − 2x − 8, and verify the same relationship.
3. Find the zeroes of 3x² + 5x − 2, and verify the relationship.
4. Find the zeroes of x² − 3 and verify the relationship.
   (Use √3 in your answer.)
5. Find a quadratic polynomial whose sum of zeroes is −3 and product of zeroes is 2.
6. If α and β are the zeroes of 2x² − 8x + 6, find:
   • α + β
   • αβ

   (Do not factorise; use coefficients.)

7. A polynomial p(x) has two distinct zeroes. What does its graph do with the x-axis?
   Answer in one line.

Advance

1. If α and β are zeroes of 4x² − 4x + 1, find α + β and αβ.
   Then decide whether the zeroes are equal or distinct.
2. Form the quadratic polynomial whose zeroes are:
   • 2 and −5

   (Take leading coefficient = 1.)

3. Form the quadratic polynomial whose zeroes are:
   • 1/3 and −2

   (Make all coefficients integers.)

4. For the cubic polynomial p(x) = 2x³ − 5x² − 14x + 8, the zeroes are 4, −2, and 1/2.
   Find:
   • α + β + γ
   • αβ + βγ + γα
   • αβγ
5. A quadratic polynomial has sum of zeroes = 5 and product of zeroes = −6.
   Write one quadratic polynomial for it.
6. A cubic polynomial has zeroes 3, −1, and −1/3.
   Write a cubic polynomial with integer coefficients and leading coefficient 3.
7. If the graph of y = p(x) touches the x-axis at exactly one point (does not cut),
   how many zeroes does p(x) have?
   Answer with reason in one line.

HOTS

1. A student says: “If a polynomial is of degree 3, it must have 3 zeroes.”
   Is this correct? Write Yes/No with one reason.
2. Without factorising, find the sum and product of zeroes of:
   p(x) = 5x² − 20x + 15
3. If α and β are zeroes of x² + px + 16 and α + β = −8, find p.
4. If α and β are zeroes of 2x² + kx + 8 and αβ = 4, find k.
5. A cubic polynomial has zeroes α, β, γ such that:
   α + β + γ = 2, αβ + βγ + γα = −7, αβγ = −4.
   Write one cubic polynomial with leading coefficient 1.
6. The graph of y = p(x) lies completely above the x-axis.
   How many real zeroes does p(x) have?
   Answer in one line.
7. If a quadratic polynomial has zeroes 2 and 2 (same zero twice),
   write its polynomial in factor form and expanded form.

Answer Key

Basic – Answers

1. Ans: The highest power of x in the polynomial is called its degree.
   Hint: Look for the greatest exponent.
2. Ans:
   • 4x + 2 → degree 1, linear
   • 5x² − 4x + 1 → degree 2, quadratic
   • 3x³ − 2x² + x − 1 → degree 3, cubic

   Hint: Degree decides the type.

3. Ans:
   • p(2) = 2² − 3×2 − 4 = 4 − 6 − 4 = −6
   • p(0) = 0 − 0 − 4 = −4
   • p(−1) = (−1)² − 3(−1) − 4 = 1 + 3 − 4 = 0

   Hint: Substitute x = given value.

4. Ans: A real number k is a zero of p(x) if p(k) = 0.
   Hint: “Zero” means value becomes 0.
5. Ans: 2x + 3 = 0 ⇒ x = −3/2.
   Hint: Set the polynomial equal to 0.
6. Ans: Maximum possible degree is 3.
   Hint: Degree n ⇒ at most n zeroes (x-axis intersections).
7. Ans:
   • p(−2) = (−2)² − 3(−2) − 4 = 4 + 6 − 4 = 6
   • p(−1) = 0
   • p(0) = −4
   • p(1) = 1 − 3 − 4 = −6
   • p(2) = −6

   Hint: Compute one-by-one using substitution.

Standard – Answers

1. Ans: x² + 7x + 10 = (x + 2)(x + 5)
   Zeroes: −2, −5
   Sum = −7 = −(7/1), Product = 10 = 10/1
   Hint: Compare with ax² + bx + c.
2. Ans: x² − 2x − 8 = (x − 4)(x + 2)
   Zeroes: 4, −2
   Sum = 2 = −(−2/1), Product = −8 = −8/1
   Hint: Split middle term or factor by inspection.
3. Ans: 3x² + 5x − 2 = (3x − 1)(x + 2)
   Zeroes: 1/3, −2
   Sum = (1/3) + (−2) = −5/3 = −(5/3)
   Product = (1/3)×(−2) = −2/3 = (−2/3)
   Hint: Use factorisation by splitting 5x.
4. Ans: x² − 3 = (x − √3)(x + √3)
   Zeroes: √3, −√3
   Sum = 0 = −(0/1), Product = −3 = −3/1
   Hint: Use a² − b² identity.
5. Ans: One polynomial is x² + 3x + 2.
   Hint: For a = 1, use x² − (sum)x + (product).
6. Ans: For 2x² − 8x + 6: a = 2, b = −8, c = 6
   α + β = −b/a = −(−8)/2 = 4
   αβ = c/a = 6/2 = 3
   Hint: Use −b/a and c/a directly.
7. Ans: It cuts the x-axis at two distinct points.
   Hint: Each x-intersection gives a zero.

Advance – Answers

1. Ans: For 4x² − 4x + 1: a = 4, b = −4, c = 1
   α + β = −b/a = 4/4 = 1
   αβ = c/a = 1/4
   Also, 4x² − 4x + 1 = (2x − 1)² ⇒ equal zeroes.
   Hint: Perfect square gives repeated root.
2. Ans: Zeroes 2 and −5 ⇒ polynomial = (x − 2)(x + 5) = x² + 3x − 10.
   Hint: Use (x − α)(x − β).
3. Ans: (x − 1/3)(x + 2) = x² + (5/3)x − 2/3
   Multiply by 3: 3x² + 5x − 2.
   Hint: Clear fractions by multiplying a constant.
4. Ans: α = 4, β = −2, γ = 1/2
   α + β + γ = 4 + (−2) + 1/2 = 2½
   αβ + βγ + γα = (4)(−2) + (−2)(1/2) + (1/2)(4) = −8 −1 +2 = −7
   αβγ = 4×(−2)×(1/2) = −4
   Hint: Compute systematically, pair-by-pair.
5. Ans: One polynomial is x² − 5x − 6.
   Hint: x² − (sum)x + (product).
6. Ans: Zeroes 3, −1, −1/3
   Polynomial with leading coefficient 1: (x − 3)(x + 1)(x + 1/3)
   Multiply by 3 to make leading coefficient 3:
   3(x − 3)(x + 1)(x + 1/3) = (x − 3)(x + 1)(3x + 1)
   Expand: (x − 3)(x + 1) = x² − 2x − 3
   (x² − 2x − 3)(3x + 1) = 3x³ − 5x² − 11x − 3
   Hint: Multiply by 3 to remove /3.
7. Ans: It has one zero (a repeated zero).
   Hint: Touching means coincident intersection point.

HOTS – Answers

1. Ans: No.
   Reason: Degree 3 means at most 3 zeroes, not necessarily 3.
   Hint: “At most” is the key phrase.
2. Ans: For 5x² − 20x + 15: a = 5, b = −20, c = 15
   Sum = −b/a = 20/5 = 4
   Product = c/a = 15/5 = 3
   Hint: Use −b/a and c/a.
3. Ans: For x² + px + 16: α + β = −p
   Given α + β = −8 ⇒ −p = −8 ⇒ p = 8
   Hint: Here a = 1, so sum = −p.
4. Ans: For 2x² + kx + 8: αβ = c/a = 8/2 = 4
   Given αβ = 4 ⇒ already satisfied for any k
   So k can be any real number.
   Hint: Product depends only on c/a.
5. Ans: For leading coefficient 1:
   p(x) = x³ − (α + β + γ)x² + (αβ + βγ + γα)x − αβγ
   = x³ − 2x² − 7x + 4
   Hint: Use the standard cubic-coefficient relationship.
6. Ans: It has 0 real zeroes.
   Hint: No x-axis intersection ⇒ no real zero.
7. Ans: Factor form: (x − 2)(x − 2) = (x − 2)²
   Expanded form: x² − 4x + 4
   Hint: Repeated zero gives a square factor.

Worksheet for Other chapters

• Real Numbers Class 10 Maths Worksheet
• Pair of Linear Equations in Two Variables Class 10 Maths Worksheet
• Quadratic Equations Class 10 Maths Worksheet
• Arithmetic Progressions Class 10 Maths Worksheet
• Introduction to Trigonometry Class 10 Maths Worksheet
• Some Applications of Trigonometry Class 10 Maths Worksheet
• Areas Related to Circles Class 10 Maths Worksheet
• Probability Class 10 Maths Worksheet