Class ten maths Chap 2 Polynomials NCERT Solution
Chapter 2: Polynomials NCERT Solutions
TEXTBOOK’S EXERCISE – 2.1
1. The graphs of y = p(x) are given in the figure below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
Sol. (i) From the graph, it is clear that it does not intersect the x-axis. Hence, it has no zero.
(ii) From the graph, it is clear that it intersects the x-axis at one point. Hence, it has one zero.
(iii) From the graph, it is clear that it intersects the x-axis at three points. Hence, it has three zeroes.
(iv) From the graph, it is clear that it intersects the x-axis at two points. Hence, it has two zeroes.
(v) From the graph, it is clear that it intersects the x-axis at four points. Hence, it has four zeroes.
(vi) From the graph, it is clear that it intersects the x-axis at three points. Hence, it has three zeroes.
TEXTBOOK’S EXERCISE – 2.2
1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x2 – 2x – 8
(ii) 4s2 – 4s + 1
(iii) 6x2 – 3 – 7x
(iv) 4u2 + 8u
(v) t2 – 15
(vi) 3x2 – x – 4
Sol. (i) Quadratic polynomial is
X2 – 2x – 8 = x2–4x + 2x – 8
=X( x–4) + 2(x–4)
=(x–4) (x–2)
The value of x2–2x – 8 is zero, if
(x – 4) = 0 or (x + 2) = 0
Then, x = 4 and x = – 2.
So, zeroes of x2 – 2x – 8 are –2 and 4
Thus, sum of zeroes = (–2) + (4) = 2
Product of zeroes = (–2) (4) = – 8
Hence, relationship between the zeroes and the coefficients is verified.
(ii) Quadratic polynomial is
4s2 – 4s+ 1= 4s2– 2s-2s+1
=2s(2s-1) – (2s-1)
= (2s-1) (2s-1)
This value of 4s2 – 4s+ 1 is zero, if (2s-1) = 0 or (2s-1)=0
Product of zeroes
Hence, relationship between the zeroes and the coefficients is verified.
(iii) Quadratic polynomial is
6x2 – 3- 7x = 6x2 – 7x -3 = 6x2 – 9x +2x – 3
=3x (2x- 3) +1(2x-3)= (2x-3) (3x +1)
The value of 6x2-3 -7x is zero, if
(2x-3) =0 or (3x+1)=0
Then x = 3/2 or x = −1/3
So, zeroes of 6x2 – 3- 7x are 3/2 and 1/3 . Thus, sum of zeroes
Hence, relationship between the zeroes and the coefficients is verified.
(iv) Quadratic polynomial is
Quadratic polynomial is 4u2 + 8u = 4u (u+2)
The value of 4u2 +8u is zero, if
if 4u = 0 or u + 2 = 0.
Then u = 0 or u = – 2.
So, zeroes of 4u2 + 8u are 0 and –2.
Thus, sum of zeroes = 0 + (–2) = – 2
Product of zeroes = (0) (–2) = 0
Hence, relationship between the zeroes and the coefficients is verified.
(v) Quadratic polynomial is
The value of t2 – 15 is zero if
Hence, relationship between the zeroes and the coefficients is verified.
(vi) Quadratic polynomial is
3x2 – x – 4 = 3x2 + 3x – 4x – 4
= 3x (x + 1) 4(x + 1)
= (x + 1) (3x – 4)
The value of 3x2 – x – 4 is zero, if
(x + 1) = 0 or 3x – 4 = 0
Then x = – 1 or x = 43
So, zeroes of 3x2 – x – 4 are –1 and 43.
Hence, the relationship between the zeroes and the coefficients is verified.