NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4 are part of NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4.

  • Polynomials Class 10 Ex 2.1
  • Polynomials Class 10 Ex 2.2
  • Polynomials Class 10 Ex 2.3
  • Polynomials Class 10 Ex 2.4
BoardCBSE
TextbookNCERT
ClassClass 10
SubjectMaths
ChapterChapter 2
Polynomials
ExerciseEx 2.4
Number of Questions Solved5
CategoryNCERT Solutions

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4

Ex 2.4 Class 10 Maths Question 1.
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case:
(i) 2x3 + x2 – 5x + 2;  \frac { 1 }{ 4 }, 1, -2
(ii) x3 – 4x2 + 5x – 2; 2, 1, 1
Solution:
(i)
 Comparing the given polynomial with ax3 + bx2 + cx + d, we get:
a = 2, b – 1, c = -5 and d = 2.
∴  p(x) = 2x3 + x2 – 5x + 2
NCERT Solutions for Class 10 Maths Chapter 2 Polynomials e4 1
NCERT Solutions for Class 10 Maths Chapter 2 Polynomials e4 1a
(ii) Compearing the given polynomial with ax3 + bx2 + cx + d, we get:
a = 1, b = -4, c = 5 and d = – 2.
∴ p (x) = x3 – 4x2 + 5x – 2
⇒  p(2) = (2)3 – 4(2)2 + 5 x 2 – 2
= 8 – 16+ 10 – 2 = 0
p(1) = (1)3 – 4(1)2 + 5 x 1-2
= 1 – 4 + 1 – 2
= 6-6 = 0
Hence, 2, 1 and 1 are the zeroes of x3 – 4x+ 5x – 2.
Hence verified.
Now we take α = 2, β = 1 and γ = 1.
α + β + γ = 2 + 1 + 1 = \frac { 4 }{ 1 } = \frac { -b }{ a }
αβ + βγ + γα = 2 + 1 + 2 = \frac { 5 }{ 1 } = \frac { c }{ a }
αβγ = 2 x 1 x 1  = \frac { 2 }{ 1 } = \frac { -d }{ a }.
Hence verified.

Ex 2.4 Class 10 Maths Question 2.
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, -7, -14 respectively.
Solution:
Let α , β and  γ be the zeroes of the required polynomial.
Then α + β + γ = 2, αβ + βγ + γα = -7 and αβγ = -14.
∴ Cubic polynomial
= x3 – (α + β + γ)x2 + (αβ + βγ + γα)x – αβγ
= x3 – 2x2 – 1x + 14
Hence, the required cubic polynomial is x3 – 2x2 – 7x + 14.

Ex 2.4 Class 10 Maths Question 3.
If the zeroes of the polynomial x3 – 3x2 + x + 1 are a-b, a, a + b, find a and b.
Solution:
Let α , β and  γ be the zeroes of polynomial x3 – 3x2 + x + 1.
Then α =  a-b, β = a and γ = a + b.
∴ Sum of zeroes = α + β + γ
⇒   3 = (a – b) + a + (a + b)
⇒  (a – b) + a + (a + b) = 3
⇒  a-b + a + a + b = 3
⇒       3a = 3
⇒ a =  \frac { 3 }{ 3 } = 1 …(i)
Product of zeroes = αβγ
⇒ -1 = (a – b) a (a + b)
⇒ (a – b) a (a + b) = -1
⇒   (a2 – b2)a = -1
⇒  a3 – ab2 = -1   … (ii)
Putting the value of a from equation (i) in equation (ii), we get:
(1)3-(1)b2 = -1
⇒ 1 – b2 = -1
⇒ – b2 = -1 – 1
⇒  b2 = 2
⇒ b = ±√2
Hence, a = 1 and b = ±√2.

Ex 2.4 Class 10 Maths Question 4.
If two zeroes of the polynomial x4 – 6x3 – 26x+ 138x – 35 are 2 ± √3, finnd other zeroes.
Solution:
Since two zeroes are 2 + √3 and 2 – √3,
∴  [x-(2 + √3)] [x- (2 – √3)]
= (x-2- √3)(x-2 + √3)
= (x-2)2– (√3)2
x2 – 4x + 1 is a factor of the given polynomial.
Now, we divide the given polynomial by x2 – 4x + 1.
study rankers class 10 maths Chapter 2 Polynomials e4 4
So, x4 – 6x3 – 26x2 + 138x – 35
= (x2 – 4x + 1) (x2 – 2x – 35)
= (x2 – 4x + 1) (x2 – 7x + 5x – 35)
= (x2-4x + 1) [x(x- 7) + 5 (x-7)]
= (x2 – 4x + 1) (x – 7) (x + 5)
Hence, the other zeroes of the given polynomial are 7 and -5.

Ex 2.4 Class 10 Maths Question 5.
If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a, find k and a.
Solution:
We have
p(x) = x4 – 6x3 + 16x2 – 25x + 10
Remainder = x + a   … (i)
Now, we divide the given polynomial 6x3 + 16x2 – 25x + 10 by x2 – 2x + k.
NCERT Solutions for Class 10 Maths Chapter 2 Polynomials e4 5
Using equation (i), we get:
(-9 + 2k)x + 10-8 k + k2 = x + a
On comparing the like coefficients, we have:
-9 + 2k = 1
⇒ 2k = 10
⇒ k = \frac { 10 }{ 2 } = 5  ….(ii)
and 10 -8k + k2– a   ….(iii)
Substituting the value of k = 5, we get:
10 – 8(5) + (5)2 = a
⇒   10 – 40 + 25 = a
⇒  35 – 40 =   a
⇒   a =   -5
Hence, k = 5 and a = -5.

We hope the NCERT Solutions for Class 10 Mathematics Chapter 2 Polynomials Ex 2.4 help you. If you have any query regarding NCERT Solutions for Class 10 Mathematics Chapter 2 Polynomials Ex 2.4 drop a comment below and we will get back to you at the earliest.

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