NCERT Solutions for Class 9 Maths Exercise 2.2 Question 3
Understanding the Question 🧐
This question asks us to verify if a given value of a variable is a “zero of the polynomial”. A value is a “zero” if, when we substitute it into the polynomial, the entire expression simplifies to 0. If the result is any other number, it is not a zero.
Verify whether the following are zeroes of the polynomial, indicated against them.
Solutions and Verifications 📝
(i) &&p(x) = 3x + 1, x = -\frac{1}{3}&&
Since the result is 0, Yes, &&-\frac{1}{3}&& is a zero of the polynomial.
(ii) &&p(x) = 5x – \pi, x = \frac{4}{5}&&
Since &&4 – \pi \neq 0&&, No, &&\frac{4}{5}&& is not a zero of the polynomial.
(iii) &&p(x) = x^2 – 1, x = 1, -1&&
For &&x = 1&&: &&p(1) = (1)^2 – 1 = 1 – 1 = 0&&.
For &&x = -1&&: &&p(-1) = (-1)^2 – 1 = 1 – 1 = 0&&.
Since both results are 0, Yes, both 1 and -1 are zeroes of the polynomial.
(iv) &&p(x) = (x + 1)(x – 2), x = -1, 2&&
For &&x = -1&&: &&p(-1) = (-1 + 1)(-1 – 2) = (0)(-3) = 0&&.
For &&x = 2&&: &&p(2) = (2 + 1)(2 – 2) = (3)(0) = 0&&.
Since both results are 0, Yes, both -1 and 2 are zeroes of the polynomial.
(v) &&p(x) = x^2, x = 0&&
Since the result is 0, Yes, 0 is a zero of the polynomial.
(vi) &&p(x) = lx + m, x = -\frac{m}{l}&&
Since the result is 0, Yes, &&-\frac{m}{l}&& is a zero of the polynomial.
(vii) &&p(x) = 3x^2 – 1, x = -\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}&&
For &&x = -\frac{1}{\sqrt{3}}&&: &&p(-\frac{1}{\sqrt{3}}) = 3(-\frac{1}{\sqrt{3}})^2 – 1 = 3(\frac{1}{3}) – 1 = 1 – 1 = 0&&.
For &&x = \frac{2}{\sqrt{3}}&&: &&p(\frac{2}{\sqrt{3}}) = 3(\frac{2}{\sqrt{3}})^2 – 1 = 3(\frac{4}{3}) – 1 = 4 – 1 = 3&&.
The first value gives 0, but the second gives 3.
Conclusion: &&-\frac{1}{\sqrt{3}}&& is a zero, but &&\frac{2}{\sqrt{3}}&& is not a zero.
(viii) &&p(x) = 2x + 1, x = \frac{1}{2}&&
Since the result is not 0, No, &&\frac{1}{2}&& is not a zero of the polynomial.
- A zero of a polynomial is a value that makes the polynomial equal to 0.
- It is also called a root of the polynomial equation &&p(x) = 0&&.
- To check if a value is a zero, simply substitute it into the polynomial. If the result is 0, it’s a zero.
FAQ ❓
Q: What is a ‘zero of a polynomial’?
A: A zero of a polynomial is a special number that, when you substitute it for the variable, makes the whole polynomial equal to 0.
Q: Can a polynomial have two zeroes?
A: Yes. A polynomial can have multiple zeroes. For example, a quadratic polynomial (degree 2) like &&x^2 – 1&& can have up to two zeroes, which we found to be 1 and -1.
Further Reading 📖
To learn more about finding the zeroes of polynomials, you can refer to the official NCERT textbook for Class 9 Maths, Chapter 2. More resources are available on the NCERT website at https://ncert.nic.in/.
