### Chapter 2: Polynomials

Q
##### Polynomials CBSE NCERT Solutions

Question:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i)                 (ii)              (iii)

(iv)                  (v)                         (vi)

(i)

Let P(x) =

Zeroes of the polynomial are the value of  where P(x) = 0.

Putting P(x) = 0 then we get

We can find zeroes of the polynomial using splitting the middle term method.

In splitting the middle term method we need to find two numbers where the sum is -2 and product is -8 x 1 = -8.

Therefore, we can write,

Either

So,  x = 4 and x = -2.

Let = 4 and  = -2 are the zeroes of the polynomial.

If we compare   with   then we get

a = 1, b = -2 and c = -8

Now we verify the relationship between the zeroes and the coefficients in two different cases.

Case I:

LHS =

RHS =

LHS = RHS

Case II:

LHS =    = 4 x (-2) = - 8

RHS =

LHS = RHS

In both cases, LHS is equal to RHS.

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

(ii)

Let P(s) =

Zeroes of the polynomial are the value of  where P(s) = 0.

Putting P(s) = 0 then we get

We can find zeroes of the polynomial using splitting the middle term method.

In splitting the middle term method we need to find two numbers where the sum is -4 and product is 4  1 = 4.

Therefore, we can write,

Let   and    are the zeroes of the polynomial.

P(s ) =

If we compare P(s) with   then we get

a = 4, b = -4 and c = 1

Now we verify the relationship between the zeroes and the coefficients in two different cases.

Case I:

LHS =

RHS =

LHS = RHS

Case II:

LHS =

RHS =

LHS = RHS

In both cases, LHS is equal to RHS.

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

(iii)

Let P(x ) =

Zeroes of the polynomial are the value of x where P(x ) = 0.

Putting P(x) = 0 then we get

We can find zeroes of the polynomial using splitting the middle term method.

In splitting the middle term method we need to find two numbers where the sum is -7 and product is 6  x (-3) = -18.

Therefore, we can write,

Either

So, x = 3/2 and x = -1/3

Let      and        are the zeroes of the polynomial.

P(x) =   =

If we compare p(x) with   then we get

a = 6, b = -7 and c = -3

Now we verify the relationship between the zeroes and the coefficients in two different cases.

Case I:

LHS =

RHS =

LHS = RHS

Case II:

LHS =

RHS =

LHS = RHS

In both cases, LHS is equal to RHS.

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

(iv)

Let P(u) =

Zeroes of the polynomial are the value of  u where P(u) = 0.

Putting P(u) = 0 then we get

We can find zeroes of the polynomial using common factor method.

4u is the common factor in the P(u) =

Therefore, we can write,

Either 4u  = 0 or  (u+2) = 0

So, u = 0 and u = -2.

Let   = 0 and  = -2 are the zeroes of the polynomial.

P(u) =

If we compare p(u) with  then we get

a = 4, b = 8 and c = 0

Now we verify the relationship between the zeroes and the coefficients in two different cases.

Case I:

LHS =

RHS =

LHS = RHS

Case II:

LHS =

RHS =

LHS = RHS

In both cases, LHS is equal to RHS.

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

(v)

Let P(t) =

Zeroes of the polynomial are the value of  t where P(t) = 0.

Putting P(t) = 0 then we get

So,   and

Let    and   =  are the zeroes of the polynomial.

P(t) =  =

If we compare p(t) with   then we get

a = 1, b = 0 and c = -15

Now we verify the relationship between the zeroes and the coefficients in two different cases.

Case I:

LHS =

RHS =

LHS = RHS

Case II:

LHS =

RHS =

LHS = RHS

In both cases, LHS is equal to RHS.

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

(vi)

Let P(x) =

Zeroes of the polynomial are the value of  where P(x) = 0.

Putting P(x) = 0 then we get

We can find zeroes of the polynomial using splitting the middle term method.

In splitting the middle term method we need to find two numbers where the sum is -1 and product is 3 x (-4) = -12.

Therefore, we can write,

Either

So,  x = 4/3 and x = -1.

Let  = 4/3 and   = -1 are the zeroes of the polynomial.

If we compare   with   then we get

a = 3, b = -1 and c = -4

Now we verify the relationship between the zeroes and the coefficients in two different cases.

Case I:

LHS =

RHS =

LHS = RHS

Case II:

LHS =

RHS =

LHS = RHS

In both cases, LHS is equal to RHS.

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

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