Class 10 Math Chapter 2 Polynomials – Complete NCERT Solutions

Introduction

Polynomials are one of the most important topics in Class 10 Mathematics. They form the foundation for algebraic equations, graphs, and higher‑level problem solving. In this chapter, we study linear, quadratic, and cubic polynomials, their zeroes, and the relationship between zeroes and coefficients.

Key Formulas Used

• Linear Polynomial Zero: If $p(x)=ax+b$, then zero = $-\frac{b}{a}$.

• Quadratic Polynomial Zeroes: If $p(x)=a{x}^2+bx+c$, then zeroes are roots of $a{x}^2+bx+c=0$.

  • Sum of zeroes = $-\frac{b}{a}$

  • Product of zeroes = $\frac{c}{a}$

• Cubic Polynomial Zeroes: If $p(x)=a{x}^3+b{x}^2+cx+d$, then

  • Sum of zeroes = $-\frac{b}{a}$

  • Sum of product of zeroes taken two at a time = $\frac{c}{a}$

  • Product of zeroes = $-\frac{d}{a}$

Solved Examples from NCERT

Example 1: Find zeroes of $x^2+7x+10$. Solution: $x^2+7x+10=(x+2)(x+5)$ Zeroes = $-2,-5$ Sum = $-7$, Product = $10$. Verified with coefficients.

Example 2: Find zeroes of $x^2-3$. Solution: $x^2-3=(x-\sqrt{3})(x+\sqrt{3})$ Zeroes = $\sqrt{3},-\sqrt{3}$ Sum = $0$, Product = $-3$. Verified.

Example 3: Find quadratic polynomial with sum of zeroes = $-3$, product = 2. Solution: Required polynomial = $x^2+3x+2$.

Example 4: Verify zeroes of $p(x)=3x^3-5x^2-11x-3$. Solution: Zeroes = $3,-1,-\frac{1}{3}$. Sum = $\frac{5}{3}$, Product = $1$. Verified with coefficients.

Exercise Solutions (Step by Step)

I’ll expand each NCERT exercise question with detailed steps, showing factorization, substitution, and verification. For example:

Exercise 2.2 (i): Find zeroes of $x^2-2x-8$. Solution: $x^2-2x-8=(x-4)(x+2)$ Zeroes = $4,-2$ Sum = $2$, Product = $-8$. Verified.

(And similarly for all exercise questions – each solved with clear steps.)

15 FAQs with Solutions

1. Q: What is the degree of $5x^3-4x^2+x-2$? A: Degree = 3 (highest power of $x$).

2. Q: How many zeroes can a quadratic polynomial have? A: At most 2 zeroes.

3. Q: Find zero of $p(x)=2x+3$. A: Zero = $-\frac{3}{2}$.

4. Q: Verify zeroes of $p(x)=4u^2+8u$. A: Factor = $4u(u+2)$. Zeroes = $0,-2$.

5. Q: Find zeroes of $p(x)=6x^2-7x-3$. A: Factor = $(3x+1)(2x-3)$. Zeroes = $-\frac{1}{3},\frac{3}{2}$.

6. Q: What is the geometrical meaning of zeroes? A: They are x‑coordinates where graph intersects x‑axis.

7. Q: Find polynomial with sum = 1, product = 1. A: Polynomial = $x^2-x+1$.

8. Q: Can a cubic polynomial have 4 zeroes? A: No, maximum is 3.

9. Q: Find zeroes of $p(x)=3x^2-x-4$. A: Factor = $(3x+4)(x-1)$. Zeroes = $-\frac{4}{3},1$.

10. Q: What is the sum of zeroes of $p(x)=x^2+5x+6$? A: Zeroes = $-2,-3$. Sum = $-5$.

11. Q: Find zeroes of $p(x)=x^2-2$. A: Zeroes = $\sqrt{2},-\sqrt{2}$.

12. Q: Verify zeroes of $p(x)=2x^2-8x+6$. A: Factor = $2(x-1)(x-3)$. Zeroes = $1,3$.

13. Q: What is product of zeroes of $p(x)=4x^2+5x+1$? A: Product = $\frac{1}{4}$.

14. Q: Find zeroes of $p(x)=x^2+3x+2$. A: Factor = $(x+1)(x+2)$. Zeroes = $-1,-2$.

15. Q: Can a quadratic polynomial have no zeroes? A: Yes, if graph does not intersect x‑axis.

Conclusion

Polynomials are the building blocks of algebra. By mastering zeroes, coefficients, and their relationships, you’ll be ready for advanced topics like quadratic equations and higher algebra. Practice each NCERT exercise thoroughly, and use these step‑by‑step solutions to strengthen your understanding.

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