Loading...

About Polynomials — Class 10 CBSE

Relate zeroes and coefficients of polynomials; perform division algorithm for polynomials. This topic is part of the CBSE Class 10 mathematics syllabus (chapter: Chapter 2). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Polynomials

  • Introduction to Polynomials: The Basics
  • Zeros of a Polynomial: Where It Crosses
  • Relationship Between Zeros and Coefficients of a Quadratic Polynomial
  • Division Algorithm for Polynomials
  • Polynomials: Putting It All Together

Interactive lesson · about 15 minutes · checkpoint question after every unit

Polynomials — solved examples for Class 10 CBSE

Example 1easy

If 'k' is a zero of the polynomial P(x) = ax² + bx + c, which of the following statements must be true?
  1. A)P(k) = 0
  2. B)P(0) = k
  3. C)P(k) = c
  4. D)P(x) = k

Step-by-step solution

  1. By definition, a number 'k' is called a zero of a polynomial P(x) if P(k) = 0.
  2. This means that when 'k' is substituted for 'x' in the polynomial, the entire expression evaluates to zero.

Answer: P(k) = 0

Example 2medium

Find the zeros of the quadratic polynomial `6x² - 7x - 3`.
  1. A)(3/2, -1/3)
  2. B)(-3/2, 1/3)
  3. C)(1/2, -3/2)
  4. D)(-1/2, 3/2)

Step-by-step solution

  1. Set the polynomial to zero: `6x² - 7x - 3 = 0`.
  2. Split the middle term: `6x² - 9x + 2x - 3 = 0`.
  3. Factor by grouping: `3x(2x - 3) + 1(2x - 3) = 0`.
  4. This gives `(2x - 3)(3x + 1) = 0`. Setting each factor to zero, `2x - 3 = 0` implies `x = 3/2`, and `3x + 1 = 0` implies `x = -1/3`.

Answer: (3/2, -1/3)

Example 3hard

If (x² + 2x + k) is a factor of (x³ + 4x² + 10x + 12), what is the value of k?
  1. A)A) 4
  2. B)B) 6
  3. C)C) 8
  4. D)D) 10

Step-by-step solution

  1. Perform polynomial long division of (x³ + 4x² + 10x + 12) by (x² + 2x + k).
  2. The first term of the quotient is x. Multiplying (x² + 2x + k) by x gives x³ + 2x² + kx. Subtract this from the dividend: (x³ + 4x² + 10x + 12) - (x³ + 2x² + kx) = 2x² + (10-k)x + 12.
  3. The next term of the quotient is 2. Multiplying (x² + 2x + k) by 2 gives 2x² + 4x + 2k. Subtract this from the current remainder: (2x² + (10-k)x + 12) - (2x² + 4x + 2k) = (10-k-4)x + (12-2k) = (6-k)x + (12-2k).
  4. For (x² + 2x + k) to be a factor, the remainder must be 0. Therefore, (6-k)x + (12-2k) = 0 for all x. This implies 6-k = 0 and 12-2k = 0. Both conditions give k = 6.

Answer: B) 6

Practice questions on Polynomials

  1. Q1.easy

    Which of the following expressions represents a quadratic polynomial?
    1. A)2x + 5
    2. B)3x² - 4x + 1
    3. C)x³ + 2x - 7
    4. D)4x⁴
    Show answer

    Answer: 3x² - 4x + 1

    Hint: Remember that the degree of a polynomial is determined by the highest power of the variable in the expression. What degree defines a quadratic polynomial?

  2. Q2.easy

    A graph of a polynomial y = P(x) is shown. If this graph intersects the x-axis at exactly three distinct points, say at x = -2, x = 1, and x = 3, what can be concluded about the number of zeros of P(x)?
    1. A)P(x) has no real zeros.
    2. B)P(x) has exactly two real zeros.
    3. C)P(x) has exactly three real zeros.
    4. D)P(x) must be a quadratic polynomial.
    Show answer

    Answer: P(x) has exactly three real zeros.

    Hint: Recall the geometric interpretation of the zeros of a polynomial. What do the points where the graph crosses or touches the x-axis represent?

  3. Q3.easy

    Ravi was finding the sum of zeros for the quadratic polynomial x² - 7x + 10. He stated that the sum of zeros is 7. Is his calculation correct? If not, what is the correct sum?
    1. A)Yes, the sum is 7.
    2. B)No, the sum is -7.
    3. C)No, the sum is 10.
    4. D)Yes, the sum is 7 because -(-7)/1 = 7.
    Show answer

    Answer: Yes, the sum is 7 because -(-7)/1 = 7.

    Hint: Remember the relationship between the zeros (α, β) and the coefficients (a, b, c) of a quadratic polynomial ax² + bx + c. What is the formula for the sum of the zeros?

  4. Q4.medium

    If one of the zeros of the quadratic polynomial `x² + 3x + k` is 2, then find the value of k.
    1. A)10
    2. B)5
    3. C)-5
    4. D)-10
    Show answer

    Answer: -10

    Hint: If a number is a zero of a polynomial, then substituting that number into the polynomial will result in zero.

  5. Q5.medium

    Find a quadratic polynomial whose sum of zeros is -3 and product of zeros is 2.
    1. A)x² + 3x + 2
    2. B)x² - 3x + 2
    3. C)x² + 3x - 2
    4. D)x² - 3x - 2
    Show answer

    Answer: x² + 3x + 2

    Hint: Recall the standard form of a quadratic polynomial in terms of the sum and product of its zeros: `x² - (sum of zeros)x + (product of zeros)`.

  6. Q6.medium

    Divide the polynomial `P(x) = x³ - 3x² + 5x - 3` by `G(x) = x² - 2` and find the quotient and remainder.
    1. A)Quotient = `x + 3`, Remainder = `7x - 9`
    2. B)Quotient = `x - 3`, Remainder = `7x - 9`
    3. C)Quotient = `x - 3`, Remainder = `9 - 7x`
    4. D)Quotient = `x + 3`, Remainder = `9 - 7x`
    Show answer

    Answer: Quotient = `x - 3`, Remainder = `7x - 9`

    Hint: Use polynomial long division. Ensure you align terms by their powers of x and subtract carefully.

  7. Q7.hard

    If α and β are the zeros of the quadratic polynomial P(x) = x² - (k+6)x + 2(2k-1), and α + β = αβ/2, then what is the value of k?
    1. A)A) 5
    2. B)B) 6
    3. C)C) 7
    4. D)D) 8
    Show answer

    Answer: C) 7

    Hint: Recall the relationships between the zeros and coefficients of a quadratic polynomial: sum of zeros = -b/a and product of zeros = c/a.

  8. Q8.hard

    A quadratic polynomial P(x) = ax² + bx + c (a ≠ 0) has a graph that opens upwards and touches the x-axis at a single point (p, 0), where p > 0. Which of the following statements must be true?
    1. A)A) b² - 4ac > 0 and a < 0
    2. B)B) b² - 4ac = 0 and a > 0
    3. C)C) b² - 4ac < 0 and c > 0
    4. D)D) b² - 4ac = 0 and b < 0
    Show answer

    Answer: B) b² - 4ac = 0 and a > 0

    Hint: Consider what 'opens upwards' tells you about 'a' and what 'touches the x-axis at a single point' tells you about the discriminant.

  9. Q9.hard

    If α and β are the zeros of the quadratic polynomial f(x) = x² - 5x + 3, find the value of α³ + β³.
    1. A)A) 65
    2. B)B) 70
    3. C)C) 75
    4. D)D) 80
    Show answer

    Answer: D) 80

    Hint: Use the sum and product of zeros, and recall the algebraic identity for α³ + β³ in terms of (α+β) and αβ.

These are 9 of the 60 questions available for Polynomials. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.