NCERT Class 10 Chapter 2 Polynomials Official CBSE Board Sample Problems Long Answer

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Question

If and are zeroes of a poIynomiaI, them form a polynomial whose zeroes are and

Solution

Let

are the zeroes of p(x)

So, sum of zeroes, and product of zeroes.

Now, and arc the zeroes of required quadratic polynomial.

So, sum of zeroes of required polynomial and product of zeroes of required polynomial

Required quadratic polynomial is given by

(Sum of zeroes) pmduct of zeroes

Question

If one zero of the quadratic polynomial is negative of the other, then find the zeroes of .

Solution

Let one zero of f(x) be a then other zero be .

So, sum of zeroes =0

Now, other given polynomial is

So, zeroes of p(x) arc -1 and -2.

Question

Find the zeroes of the quadratic polynomial and verify the relationship between the zeroes and the coefficients.

Solution

p(x)

So, zeroes of p(x) are and .

Now,

Sum of zeroes and product of zeroes

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

Question

Obtain all other zeroes of the polynomial, ii two of its zeroes are and.

Solution

Consider

It is given that and are zeroes of polynomial f(x)

is zero of polynomial f(x)

is factor of polynomial fix) -

Similarly, -2 is a zero of polynomial f(x).

is a factor of polynomial f(x).

Hence is a factor of polynomial f(x)

is a factor of polynomial f(x).

F(x) is divisible by -

Now

For other zeroes

Put.

Put

or

Other zeroes are and

Division algorithm

Division Algorithm

Question

On dividing by a polynomial g(x), the quotient and remainder were and , respectively. Find g(x).

Solution

Question

Obtain all the zeros of the polynomial, . if two of its zeros are and

Solution

Let p(y) =

Two of its zeroes are and

So is a factor of p(y) after long division

=

All the zeroes are,-1 and -1

Question

Find the zeroes of the polynomial

. Hence verify the relationship between the zeroes and the coefficients

Solution

Zeroes of the polynomial are and

Now and

Hence verified

also

Hence verified

Question

Divide by and verify the division algorithm.

Solution

Given f(x) .

Given g(x)

We need to divide f(x) by g(x).

Division algorithm

Division Algorithm

Here,

Dividend

Divisor

Quotient

Remainder = -5

Division Algorithm:

Dividend

LHS = RHS

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