## 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* − *b*, *a* + *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, = *x*^{2} + 4 − 4*x − *3

*= x*^{2} − 4*x + *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 *x*^{2} − 4*x + *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