Class 9 Maths Chapter 2 Polynomials – Exercise 2.3 NCERT Solutions

Introduction

Exercise 2.3 introduces the division algorithm for polynomials. You’ll learn how to divide one polynomial by another, find quotient and remainder, and verify standard identities.

Key Formula Used

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

$$p(x)=g(x)\cdot q(x)+r(x)$$

where $q(x)$ is the quotient and $r(x)$ is the remainder, and degree of $r(x)$ < degree of $g(x)$.

Solved Questions (Step by Step)

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

Solution:

• $x^2+3x+2=(x+1)(x+2)$.

• Quotient = $x+2$, Remainder = 0.

Q2. Divide $x^2-4$ by $x-2$.

Solution:

• $x^2-4=(x-2)(x+2)$.

• Quotient = $x+2$, Remainder = 0.

Q3. Divide $x^3-1$ by $x-1$.

Solution:

• $x^3-1=(x-1)(x^2+x+1)$.

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

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

Solution:

• $x^3+3x^2+3x+1=(x+1{)}^3$.

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

Q5. Divide $x^2+5x+6$ by $x+2$.

Solution:

• $x^2+5x+6=(x+2)(x+3)$.

• Quotient = $x+3$, Remainder = 0.

Q6. Divide $x^2-2x+1$ by $x-1$.

Solution:

• $x^2-2x+1=(x-1{)}^2$.

• Quotient = $x-1$, Remainder = 0.

Q7. Divide $x^3-8$ by $x-2$.

Solution:

• $x^3-8=(x-2)(x^2+2x+4)$.

• Quotient = $x^2+2x+4$, Remainder = 0.

Q8. Divide $x^3+27$ by $x+3$.

Solution:

• $x^3+27=(x+3)(x^2-3x+9)$.

• Quotient = $x^2-3x+9$, Remainder = 0.

Q9. Divide $x^4-1$ by $x-1$.

Solution:

• $x^4-1=(x-1)(x^3+x^2+x+1)$.

• Quotient = $x^3+x^2+x+1$, Remainder = 0.

Q10. Divide $x^2+4x+4$ by $x+2$.

Solution:

• $x^2+4x+4=(x+2{)}^2$.

• Quotient = $x+2$, Remainder = 0.

Q11. Divide $x^3-6x^2+11x-6$ by $x-1$.

Solution:

• $x^3-6x^2+11x-6=(x-1)(x^2-5x+6)$.

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

Q12. Divide $x^3-3x^2+3x-1$ by $x-1$.

Solution:

• $x^3-3x^2+3x-1=(x-1{)}^3$.

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

Q13. Divide $x^2-9$ by $x-3$.

Solution:

• $x^2-9=(x-3)(x+3)$.

• Quotient = $x+3$, Remainder = 0.

Q14. Divide $x^3+x^2+x+1$ by $x+1$.

Solution:

• $x^3+x^2+x+1=(x+1)(x^2+1)$.

• Quotient = $x^2+1$, Remainder = 0.

Q15. Divide $x^4+4x^2+4$ by $x^2+2$.

Solution:

• $x^4+4x^2+4=(x^2+2{)}^2$.

• Quotient = $x^2+2$, Remainder = 0.

FAQs (10 for Exercise 2.3)

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

2. Q: When is the remainder zero? A: When the divisor is a factor of the dividend polynomial.

3. Q: How do you check if a polynomial is divisible by another? A: Perform division; if remainder = 0, it’s divisible.

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

5. Q: Why is polynomial division important? A: It helps in factorization and solving polynomial equations.

6. Q: Can polynomials always be divided? A: Yes, but remainder may not be zero.

7. Q: What is the degree of remainder in division algorithm? A: Less than the degree of divisor.

8. Q: What happens if divisor is linear? A: Remainder will be a constant.

9. Q: What is synthetic division? A: A shortcut method for dividing polynomials by linear divisors.

10. Q: Are polynomial identities always true? A: Yes, they hold for all values of variables.

Conclusion

Exercise 2.3 has 15 questions that strengthen your understanding of the division algorithm and polynomial identities. These skills are essential for factorization and solving higher‑degree equations.

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