Polynomials are one of the most important topics in mathematics and are part of the CBSE syllabus in Class 10. Polynomials Class 10 is a very important part of the curriculum as it forms the basis for various other topics such as quadratic equations, factorization, and algebraic identities. In this article, we will study Polynomials Class 10 in-depth and cover all the essential concepts that you need to know along with Polynomials Class 10 extra questions and Polynomials Class 10 MCQ.

## Context

## Polynomials Class 10 Notes

### Define Polynomial.

A **polynomial **is a mathematical expression consisting of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

Examples of polynomials include quadratic expressions like x^{2} + 2x + 1, and cubic expressions like 2x^{3} – 3x^{2 }+ x – 4.

### Define Quadratic Polynomial.

A **quadratic polynomial** is a type of polynomial with a degree of 2, meaning it has the highest power of the variable of 2. It is also known as a second-degree polynomial. The general form of a quadratic polynomial is ax^{2} + bx + c, where a, b, and c are constants, and x is the variable.

### Define Cubic Polynomial.

A **cubic polynomial** is a polynomial of degree three, which means it has the highest power of the variable of 3. The general form of a cubic polynomial is ax^{3} + bx^{2} + cx + d, where a, b, c, and d are constants, and x is the variable.

## Polynomials Class 10 Extra Questions

1. Find a quadratic polynomial, whose zeroes are -4 and -5.

2. Form a quadratic polynomial whose zeroes are 3 + √2 and 3 – √2.

3. Find a quadratic polynomial, the sum, and the product of whose zeroes are √3 and 1/√3 respectively.

4. Find a quadratic polynomial, the sum and product of whose zeroes are 0 and -√2 respectively.

5. Find the zeroes of the quadratic polynomial √3 x^{2} – 8x + 4√3.

6. Find the quadratic polynomial whose zeroes are -2 and -5. Verify the relationship between zeroes and coefficients of the polynomial.

7. Find the zeroes of the quadratic polynomial 3x^{2} – 75 and verify the relationship between the zeroes and the coefficients.

8. Find the zeroes of p(x) = 2x^{2} – x – 6 and verify the relationship of zeroes with these coefficients.

9. Verify whether 2, 3, and 1/2 are the zeroes of the polynomial p(x) = 2x^{3} – 11x^{2} + 17x – 6.

10. Show that 1/2 and −3/2 are the zeroes of the polynomial 4x^{2} + 4x – 3 and verify the relationship between zeroes and coefficients of polynomials.

11. Find a quadratic polynomial, the sum and product of whose zeroes are -8 and 12 respectively. Hence find the zeroes.

12. Find a quadratic polynomial, the sum, and the product of whose zeroes are 0 and −35/35 respectively. Hence find the zeroes.

13. Find the zeroes of the quadratic polynomial 6x^{2} – 3 – 7x and verify the relationship between the zeroes and the coefficients of the polynomial.

14. Find the zeroes of the quadratic polynomial f(x) = x^{2} – 3x – 28 and verify the relationship between the zeroes and the coefficients of the polynomial

15. If a polynomial x^{4} + 5x^{3} + 4x^{2} – 10x – 12 has two zeroes as -2 and -3, then find the other zeroes.

16. Find all the zeroes of the polynomial 8x^{4} + 8x^{3} – 18x^{2} – 20x – 5, if it is given that two of its zeroes are √5/2 and −√5/2.

17. If the sum of zeroes of the quadratic polynomial 3x^{2} – kx + 6 is 3, then find the value of k.

18. If α and β are the zeroes of the polynomial ax^{2} + bx + c, find the value of α^{2} + β^{2}.

19. If the sum of the zeroes of the polynomial p(x) = (k^{2} – 14) x^{2} – 2x – 12 is 1, then find the value of k.

20. If α and β are the zeroes of a polynomial such that α + β = -6 and αβ = 5, then find the polynomial.

21. Find the condition that zeroes of polynomial p(x) = ax^{2} + bx + c are reciprocal of each other.

22. If the zeroes of the polynomial x^{2} + px + q are double in value to the zeroes of 2x^{2} – 5x – 3, find the value of p and q.

23. If α and β are the zeroes of the polynomial 6y^{2} – 7y + 2, find a quadratic polynomial whose zeroes are 1/α and 1/β.

24. If α and β are zeroes of p(x) = kx^{2} + 4x + 4, such that α^{2} + β^{2} = 24, find k.

25. If α and β are the zeroes of the polynomial p(x) = 2x^{2} + 5x + k, satisfying the relation, α^{2} + β^{2} + αβ = 21/4 then find the value of k.

26. Can (x – 2) be the remainder on division of a polynomial p(x) by (2x + 3)? Justify your answer.27.

27. What must be subtracted from the polynomial f(x) = x^{4} + 2x^{3} – 13x^{2} – 12x + 21 so that the resulting polynomial is exactly divisible by x^{2} – 4x + 3?

28. Divide 3x^{2} + 5x – 1 by x + 2 and verify the division algorithm.

29. On dividing 3x^{3} + 4x^{2} + 5x – 13 by a polynomial g(x) the quotient and remainder were 3x +10 and 16x – 43 respectively. Find the polynomial g(x).

30. Check whether polynomial x – 1 is a factor of the polynomial x^{3} – 8x^{2} + 19x – 12. Verify by division algorithm.

31. Divide 4x^{3} + 2x^{2} + 5x – 6 by 2x^{2} + 1 + 3x and verify the division algorithm.

32. Given that x – √5 is a factor of the polynomial x^{3} – 3√5 x^{2} – 5x + 15√5, find all the zeroes of the polynomial.

33. If a polynomial x^{4} + 5x^{3} + 4x^{2} – 10x – 12 has two zeroes as -2 and -3, then find the other zeroes.

34. If p(x) = x^{3} – 2x^{2} + kx + 5 is divided by (x – 2), the remainder is 11. Find k. Hence find all the zeroes of x^{3} + kx^{2} + 3x + 1.

35. What must be subtracted from p(x) = 8x^{4} + 14x^{3} – 2x^{2} + 8x – 12 so that 4x^{2} + 3x – 2 is factor of p(x)? This question was given to a group of students for working together.

36. Find the values of a and b so that x^{4} + x^{3} + 8x^{2} +ax – b is divisible by x^{2} + 1.

37. If a polynomial 3x^{4} – 4x^{3} – 16x^{2} + 15x + 14 is divided by another polynomial x^{2} – 4, the remainder comes out to be px + q. Find the value of p and q.

38. If the polynomial (x^{4} + 2x^{3} + 8x^{2} + 12x + 18) is divided by another polynomial (x^{2} + 5), the remainder comes out to be (px + q), find the values of p and q.

39. Divide the polynomial f(x) = 3x^{2} – x^{3} – 3x + 5 by the polynomial g(x) = x – 1 – x^{2} and verify the division algorithm.

40. Find a quadratic polynomial whose zeroes are reciprocals of the zeroes of the polynomial f(x) = ax^{2} + bx + c, a ≠ 0, c ≠ 0.

## Polynomials Class 10 MCQ

**1. If one zero of the quadratic polynomial x² + 3x + k is 2, then the value of k is**

(a) 10

(b) -10

(c) 5

(d) -5

**2. A quadratic polynomial, whose zeroes are -3 and 4, is**

(a) x²- x + 12

(b) x² + x + 12

(c) x^{2}/2−x/2−6

(d) 2x² + 2x – 24

**3. What is the quadratic polynomial whose sum and the product of zeroes are √2, ⅓ respectively?**

(a) 3×2-3√2x+1

(b) 3×2+3√2x+1

(c) 3×2+3√2x-1

(d) None of the above

**4. The degree of the polynomial, x ^{4} – x^{2} +2 is**

(a) 2

(b) 4

(c) 1

(d) 0

**5. If p(x) is a polynomial of degree one and p(a) = 0, then a is said to be:**

(a) Zero of p(x)

(b) Value of p(x)

(c) Constant of p(x)

(d) None of the above

**6. Zeroes of a polynomial can be expressed graphically. Number of zeroes of polynomial is equal to the number of points where the graph of the polynomial is:**

(a) Intersects x-axis

(b) Intersects y-axis

(c) Intersects y-axis or x-axis

(d) None of the above

**7. Given that two of the zeroes of the cubic polynomial ax ^{3} + bx^{2} + cx + d are 0, the third zero is**

(a) -b/a

(b) b/a

(c) c/a

(d) -d/a

**8. If one zero of the quadratic polynomial x ^{2} + 3x + k is 2, then the value of k is**

(a) 10

(b) –10

(c) 5

(d) –5

**9. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and -3, then**

(a) a = -7, b = -1

(b) a = 5, b = -1

(c) a = 2, b = -6

(d) a – 0, b = -6

**10. If one of the zeroes of a quadratic polynomial of form x² + ax + b is the negative of the other, then it**

(a) has no linear term and the constant term is negative.

(b) has no linear term and the constant term is positive.

(c) can have a linear term but the constant term is negative.

(d) can have a linear term but the constant term is positive.

**11. The zeroes of the quadratic polynomial x² + 1750x + 175000 are**

(a) both positive

(b) one positive and one negative

(c) both negative

(d) both equal

**12. If the zeroes of the quadratic polynomial Ax² + Bx + C, C # 0 are equal, then**

(a) A and B have the same sign

(b) A and C have the same sign

(c) B and C have the same sign

(d) A and C have opposite signs

**13. If x ^{3} + 1 is divided by x² + 5, then the possible degree of quotient is**

(a) 0

(b) 1

(c) 2

(d) 3

14. If one of the zeroes of the cubic polynomial x^{3} + px² + qx + r is -1, then the product of the other two zeroes is

(a) p + q + 1

(b) p-q- 1

(c) q – p + 1

(d) q – p – 1

**15. If one zero of the quadratic polynomial x² + 3x + b is 2, then the value of b is**

(a) 10

(b) -8

(c) 9

(d) -10

**16. If 1 is one of the zeroes of the polynomial x² + x + k, then the value of k is:**

(a) 2

(b) -2

(c) 4

(d) -4

**17. The zeroes of the quadratic polynomial x ^{2} + 7x + 10 are**

(a) -2, -5

(b) 2, 5

(c) -4, -3

(d) -2, 5

**18. If the discriminant of a quadratic polynomial, D > 0, then the polynomial has**

(a) two real and equal roots

(b) two real and unequal roots

(c) imaginary roots

(d) no roots

**19. If on the division of a polynomial p(x) by a polynomial g(x), the quotient is zero, then the relation between the degrees of p(x) and g(x) is**

(a) degree of p(x) < degree of g(x)

(b) degree of p(x) = degree of g(x)

(c) degree of p(x) > degree of g(x)

(d) nothing can be said about degrees of p(x) and g(x)

**20. If the graph of a polynomial intersects the x-axis at three points, then it contains ____ zeroes.**

(a) Three

(b) Two

(c) Four

(d) More than three

We hope Polynomials class 10 extra questions and notes will help you a lot while learning polynomials class 10. Also the polynomials class 10 MCQ is very important for the CBSE Board examination. If you have any queries regarding polynomial class 10, please let us know in the comment section.