Q. 14.2( 62 Votes )

# Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Answer :

(i)

factorize the given polynomial by splitting the middle term:

⇒ x^{2} - 4x + 2x – 8

⇒ x (x - 4) + 2 (x - 4)

For zeros of f(x),

f(x) = 0

⇒(x + 2) (x - 4) = 0

x+2=0

x=-2

x-4=0

x=4

⇒x = -2, 4

Therefore zeros of the polynomial are -2 & 4

In a polynomial the relations hold are as follows:

sum of zeroes is equal to

product of zeroes is equal to

For the given polynomial,

Sum of zeros = -2 + 4 = 2

And is -(-2) = 2

Hence the value of and sum of zeroes are same.

Product of zeros = -2 × 4 = -8

is -8.

Hence the value of and product of zeroes are same.

(ii)

factorize the given polynomial by splitting the middle term:

⇒ 4s^{2} -2s - 2s + 1

⇒ 2s (2s - 1) -1 (2s - 1)

For zeros of g(s), g(s) = 0

(2s - 1) (2s - 1) = 0

2s - 1=0

s =

Therefore zeros of the polynomial are ,

In a polynomial the relations hold are as follows:

sum of zeroes is equal to

product of zeroes is equal to

For the given polynomial,

Sum of zeros = + = 1

Hence the value of and sum of zeroes are same.

Product of zeros = × =

Hence the value of and product of zeroes are same.

(iii)

use the formula to solve the above equation,^{Here a is t and b is .Solve the given expression as:}

For zeros of h(t),

h(t) = 0

Therefore zeros of the given polynomial are t = √15 & -√15

In a polynomial the relations hold are as follows:

sum of zeroes is equal to

product of zeroes is equal to

For the given polynomial,

Sum of zeros = √15 + (- √15) = 0

The value of is 0.

Hence, the value of and sum of zeroes are same.

Product of zeros

The value of is -^{. Hence the value of and product of zeroes are same.}

(iv) f(x) =

Write the equation in the form of ax^{2} +bx+c as:

6x^{2} - 7x -3

factorize the given polynomial by splitting the middle term:

⇒ 6x^{2} - 9x + 2x - 3

⇒ 3x(2x - 3) +1(2x - 3)

⇒ (3x + 1) (2x - 3)

For zeros of f(x),

f(x) = 0

⇒ (3x + 1) (2x - 3) = 0

x =

Therefore zeros of the polynomial are

In a polynomial the relations hold are as follows:

sum of zeroes is equal to

product of zeroes is equal to

Sum of zeros = = = = =

Product of zeros = × = = =

(v)

P (x) = x^{2} + 3√2x - √2x - 6

For zeros of p(x), p(x) = 0

⇒ x (x + 3√2) -√2 (x + 3√2) = 0

⇒ (x - √2) (x + 3√2) = 0

x = √2, -3√2

Therefore zeros of the polynomial are √2 & -3√2

Sum of zeros = √2 -3√2 = -2√2 = -2√2 =

Product of zeros = √2 × -3√2 = -6 = -6 =

(vi) q (x) = √3x^{2} + 10x + 7√3

⇒ √3x^{2} + 10x + 7√3

⇒ √3x^{2} + 7x + 3x + 7√3

⇒ √3x (x +) + 3 (x + )

⇒ (√3x + 3) (x +)

For zeros of Q(x), Q(x) = 0

(√3x + 3) (x +) = 0

X = ,

Therefore zeros of the polynomial are ,

Sum of zeros = + =

Product of zeros = = × = 7 =

(vii) f(x) = x^{2} - (√3 + 1)x + √3

f(x) = x^{2} - √3x - x + √3

f(x) = x(x - √3) -1(x - √3)

f(x) = (x - 1) (x - √3)

For zeros of f(x), f(x) = 0

(x - 1) (x - √3) = 0

X = 1, √3

Therefore zeros of the polynomial are 1 & √3

Sum of zeros = 1 + √3 = √3 + 1=

Product of zeros = 1 × √3 = √3=

(viii) g(x) = a(x^{2} + 13) – x(a^{2} + 1)

g(x) = ax^{2} - a^{2}x – x + a

g(x) = ax^{2} - (a^{2} + 1)x + a

g(x) = ax(x - a) -1(x - a)

g(x) = (ax - 1) (x - a)

For zeros of g(x), g(x) = 0

(ax - 1) (x - a) = 0

X = , a

Therefore zeros of the polynomial are & a

Sum of zeros

Product of zeros = × a = 1 = 1 =

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