CBSE Class 10 - Maths - Polynomials - NCERT Ex 2.1

POLYNOMIALS

NCERT Exercise 2.1 and other Q & A

Q1: The graphs of y = p(x) are given in below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Answer: 

(i) As shown in the graph, it does not intersect x-axis. Hence it has no real zeroes.

(ii) From the graph, it intersects x-axis at one point. ∴ it has one real zero.

(iii) As the graph shows, it intersects x-axis at three points. ∴ the graph polynomial has three real zeroes.

(iv) From the graph, it is clear it intersects x-axis at two points. Hence it has two zeroes.

(v) As the graph shows, it meets x-axis at four points. ∴ it has four real zeroes.

(vi) As shown in the graph, it intersects x-axis at three points. ∴ it has three zeroes.

Q2(CBSE 2011): In the figure below the number of zeroes of y = f(x) are:

(a) 1

(b) 2

(c) 3

(d) 4

Answer: (c) 3. Since the graph intersects x-axis at three points.

Q3(CBSE 2011): The number of zeroes for the polynomial  y = p(x) from the given graph is :

(a) 3

(b) 1

(c) 2

(d) 0

Answer: (b) 1

Q4: Draw the graph of the quadratic polynomial p(x) = x² +6x + 9 and find zeroes of it.

Answer: Given p(x) = x² + 6x + 9

To find the zeroes of p(x), consider p(x) = 0   [Factor theorem]

∴ x² + 6x + 9 = 0

⇒ (x + 3)² = 0

⇒ x = -3.

∴ -3 is the zero of the polynomial.

For the graph, we take the following different values of x and y = p(x) = x² + 6x + 9

x	-3	-2	-1	-4
y=p(x)	0	1	4	1

The graph plotted is shown below:

Comparing the quadratic polynomial as ax² + bx + c, ∵ a =1 > 0, the graph parabola opens upwards(∪).