Class 10 Maths Chapter 2 Polynomials – Exercise 2.2 NCERT Solutions

Introduction

Exercise 2.2 focuses on the division algorithm for polynomials and the Remainder Theorem. You’ll learn how to divide polynomials, find quotient and remainder, and verify results using the theorem. This exercise builds the foundation for polynomial equations and factorization.

Formula Used

• Division Algorithm for Polynomials: If $f(x)$ and $g(x)$ are polynomials with $g(x)\mathrm{\ne }0$, then:

$$f(x)=g(x)\cdot q(x)+r(x),\text{where }r(x)=0\text{ or degree of }r(x)<\text{degree of }g(x)$$

• Remainder Theorem: If $f(x)$ is divided by $(x-a)$, then remainder = $f(a)$.

NCERT Questions with Solutions (10)

Q1. Divide $x^3-3x^2+x+1$ by $x-2$.

Solution: Using division algorithm: Quotient = $x^2-x-1$, Remainder = $-1$.

Q2. Divide $x^3+x^2+x+1$ by $x+1$.

Solution: Quotient = $x^2$, Remainder = 1.

Q3. Divide $x^3-4x^2+5x-2$ by $x-2$.

Solution: Quotient = $x^2-2x+1$, Remainder = 0.

Q4. Divide $x^3+3x^2+3x+1$ by $x+1$.

Solution: Quotient = $x^2+2x+1$, Remainder = 0.

Q5. Divide $x^3-6x^2+11x-6$ by $x-1$.

Solution: Quotient = $x^2-5x+6$, Remainder = 0.

Q6. Divide $x^3-7x+6$ by $x-2$.

Solution: Quotient = $x^2+2x-3$, Remainder = 0.

Q7. Divide $x^3+2x^2-5x+6$ by $x+2$.

Solution: Quotient = $x^2-5$, Remainder = 16.

Q8. Divide $x^3-2x^2-x+2$ by $x-1$.

Solution: Quotient = $x^2-x-2$, Remainder = 0.

Q9. Divide $x^3+4x^2+5x+2$ by $x+2$.

Solution: Quotient = $x^2+2x+1$, Remainder = 0.

Q10. Divide $x^3-3x+2$ by $x-1$.

Solution: Quotient = $x^2+x-2$, Remainder = 0.

FAQs (10 from NCERT)

1. Q: What is the division algorithm for polynomials? A: It states $f(x)=g(x)q(x)+r(x)$.

2. Q: What is the Remainder Theorem? A: If $f(x)$ is divided by $(x-a)$, remainder = $f(a)$.

3. Q: What is a quotient in polynomial division? A: The result obtained after dividing.

4. Q: What is a remainder in polynomial division? A: The leftover part after division.

5. Q: When is remainder zero? A: When divisor is a factor of dividend.

6. Q: What is a factor theorem? A: If $f(a)=0$, then $(x-a)$ is a factor of $f(x)$.

7. Q: How is remainder theorem useful? A: It helps in checking factors quickly.

8. Q: What is degree of a polynomial? A: Highest power of variable in polynomial.

9. Q: What is a linear polynomial? A: A polynomial of degree 1.

10. Q: Why is this exercise important? A: It builds the foundation for factorization and solving polynomial equations.

Conclusion

Exercise 2.2 has 10 solved questions and 10 FAQs that strengthen your understanding of polynomial division and remainder theorem. This builds the foundation for factorization and higher‑order polynomial problems in Class 10 Maths.

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