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NCERT solution class 10 chapter 2 polynomials exercise 2.4 mathematics

EXERCISE 2.4


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:

Answer:

(i) 

Therefore, , 1, and −2 are the zeroes of the given polynomial.

Comparing the given polynomial with , we obtain a = 2, b = 1, c = −5, d = 2

Therefore, the relationship between the zeroes and the coefficients is verified.

(ii) 

Therefore, 2, 1, 1 are the zeroes of the given polynomial.

Comparing the given polynomial with , we obtain a = 1, b = −4, c = 5, d = −2.

Verification of the relationship between zeroes and coefficient of the given polynomial

Multiplication of zeroes taking two at a time = (2)(1) + (1)(1) + (2)(1) =2 + 1 + 2 = 5 

Multiplication of zeroes = 2 × 1 × 1 = 2 

Hence, the relationship between the zeroes and the coefficients is verified.

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.

Answer:

Let the polynomial be and the zeroes be .

It is given that

If a = 1, then b = −2, c = −7, d = 14

Hence, the polynomial is .


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Question 3:

If the zeroes of polynomial  are, find a and b.

Answer:

Zeroes are a − ba + a + b

Comparing the given polynomial with , we obtain

p = 1, q = −3, r = 1, t = 1

The zeroes are .

Hence, a = 1 and b =  or .


Question 4:

]It two zeroes of the polynomial  are, find other zeroes.

Answer:

Given that 2 + and 2­­ are zeroes of the given polynomial.

Therefore, x2 + 4 ­­− 4x − 3

= x2 ­− 4x + 1 is a factor of the given polynomial

For finding the remaining zeroes of the given polynomial, we will find the quotient by dividing  by x2 ­− 4x + 1.

Clearly, = 

It can be observed that is also a factor of the given polynomial.

And 

Therefore, the value of the polynomial is also zero when or 

Or x = 7 or −5

Hence, 7 and −5 are also zeroes of this polynomial.


Question 5:

If the polynomial  is divided by another polynomial, the remainder comes out to be x + a, find k and a.

Answer:

By division algorithm,

Dividend = Divisor × Quotient + Remainder

Dividend − Remainder = Divisor × Quotient

 will be perfectly divisible by .

Let us divide  by 

It can be observed that will be 0.

Therefore, = 0 and = 0

For = 0,

2 k =10

And thus, k = 5

For = 0

10 − a − 8 × 5 + 25 = 0

10 − a − 40 + 25 = 0

− 5 − a = 0

Therefore, a = −5

Hence, k = 5 and a = −5


 

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