Class 10 Maths Chapter 2 Polynomials – Exercise 2.3 NCERT Solutions

Introduction

Exercise 2.3 focuses on zeros of polynomials and the relationship between coefficients and zeros. You’ll learn how to find zeros of quadratic and cubic polynomials, verify them using the factor theorem, and apply the relation between coefficients and roots.

Formula Used

• Quadratic Polynomial: For $a{x}^2+bx+c$, zeros are:

$$\alpha ,\beta =\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

• Relation between coefficients and zeros:

$$\alpha +\beta =-\frac{b}{a},\alpha \cdot \beta =\frac{c}{a}$$

• Factor Theorem: If $f(a)=0$, then $(x-a)$ is a factor of $f(x)$.

NCERT Questions with Solutions (10)

Q1. Find the zeros of $x^2-7x+10$.

Solution: Factorize: $x^2-7x+10=(x-5)(x-2)$. Zeros = 5, 2.

Q2. Find the zeros of $x^2-4x+4$.

Solution: Factorize: $x^2-4x+4=(x-2{)}^2$. Zeros = 2, 2.

Q3. Find the zeros of $x^2+5x+6$.

Solution: Factorize: $x^2+5x+6=(x+2)(x+3)$. Zeros = -2, -3.

Q4. Find the zeros of $x^2-3x-10$.

Solution: Factorize: $x^2-3x-10=(x-5)(x+2)$. Zeros = 5, -2.

Q5. Find the zeros of $x^2+7x+12$.

Solution: Factorize: $x^2+7x+12=(x+3)(x+4)$. Zeros = -3, -4.

Q6. Find the zeros of $x^2-8x+15$.

Solution: Factorize: $x^2-8x+15=(x-3)(x-5)$. Zeros = 3, 5.

Q7. Find the zeros of $x^2+x-6$.

Solution: Factorize: $x^2+x-6=(x+3)(x-2)$. Zeros = -3, 2.

Q8. Find the zeros of $x^2-9$.

Solution: Factorize: $x^2-9=(x-3)(x+3)$. Zeros = 3, -3.

Q9. Find the zeros of $x^2+2x-15$.

Solution: Factorize: $x^2+2x-15=(x+5)(x-3)$. Zeros = -5, 3.

Q10. Verify the relationship between zeros and coefficients for $x^2-5x+6$.

Solution: Factorize: $x^2-5x+6=(x-2)(x-3)$. Zeros = 2, 3. $\alpha +\beta =2+3=5=-\frac{b}{a}$. $\alpha \cdot \beta =2\cdot 3=6=\frac{c}{a}$. Verified.

FAQs (10 from NCERT)

1. Q: What is a zero of a polynomial? A: A value of variable that makes the polynomial equal to zero.

2. Q: What is the relation between zeros and coefficients of a quadratic polynomial? A: $\alpha +\beta =-\frac{b}{a},\alpha \cdot \beta =\frac{c}{a}$.

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

4. Q: What is a quadratic polynomial? A: A polynomial of degree 2.

5. Q: How do you find zeros of a quadratic polynomial? A: By factorization or quadratic formula.

6. Q: What is the discriminant of a quadratic polynomial? A: $D=b^2-4ac$.

7. Q: What does discriminant tell us? A: Nature of roots (real, equal, or complex).

8. Q: What is a cubic polynomial? A: A polynomial of degree 3.

9. Q: Can a quadratic polynomial have equal zeros? A: Yes, if discriminant = 0.

10. Q: Why is this exercise important? A: It builds understanding of zeros, coefficients, and factorization.

Conclusion

Exercise 2.3 has 10 solved questions and 10 FAQs that strengthen your understanding of zeros of polynomials and their relation with coefficients. This builds the foundation for solving polynomial equations in Class 10 Maths.

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