Polynomials

NCERT Exercise 2.2

Question 1: Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) `x^2 – 2x – 8`

Answer: `x^2 – 2x – 8 = 0`
Or, `x^2 – 4x + 2x – 8 = 0`
Or, `x(x – 4) + 2(x – 4) = 0`
Or, `(x + 2)(x – 4) = 0`

Hence, zeroes are -2 and 4

We know that; sum of zeroes `= - b/a`
Or, `- 2 + 4 = - (-2)`
Or, `text(LHS) = text(RHS)`

Again we know that; product of zeroes `= c/a`

Or, `- 2 xx 4 = - 8`

From sum and product of zeroes, the relationship between the zeroes and coefficients is verified.

(ii) `4s^2 – 4s + 1`

Answer: `4s^2 – 4s + 1 = 0`
Or, `4s^2 – 2s – 2s + 1 = 0`
Or, `4s(s – ½ ) – 2(s – ½ ) = 0`
Or, `(4s – 2)(s – ½) = 0`

Here, zeroes are; ½

We know that; sum of zeroes `= - b/a`
Or, `½ + ½ = - (- 4/4)`
Or, `1 = 1`

We know that; product of zeroes `= c/a`
Or, `½ xx ½ = ¼`

From sum and product of zeroes, the relationship between the zeroes and coefficients is verified.

(iii) `6x^2 – 3 – 7x`

Answer: `6x^2 – 7x – 3 = 0`
Or, `6x^2 - 9x + 2x – 3 = 0`
Or, `3x (2x – 3) + 1(2x – 3) = 0`
Or, `(3x + 1) (2x – 3) = 0`

Here, zeroes are; - ½ and 3/2

We know that; sum of zeroes `= - b/a`
Or, `1/2+1/2=-(-4/4)`
Or, `1=1`

We know that products of zeroes `=c/a`
Or, `1/2xx1/2=1/4`

From sum and product of zeroes, the relationship between the zeroes and coefficients is verified.

(iv) `4u^2 + 8u`

Answer: `4u^2 + 8u = 0`

Or, `u^2 + 2u = 0`

Or, `u(u + 2) = 0`

Hence, zeroes are; 0 and – 2

We know that; sum of zeroes `= - b/a`

Or, `0 – 2 = - 2`

We know that; product of zeroes `= c/a`

Or, `0 xx (- 2) = 0`

From sum and product of zeroes, the relationship between the zeroes and coefficients is verified

(v) `t^2 – 15`

Answer: `t^2 – 15 = 0`
Or, `t^2 = 15`
Or, `t = sqrt15`

Hence, zeroes are `±sqrt15`

We know that; sum of zeroes `= - b/a`

Or, `- sqrt15 + sqrt15 = 0`

We know that; product of zeroes `= c/a`

Or, `sqrt15 xx\ sqrt15 = 15`

From sum and product of zeroes, the relationship between the zeroes and coefficients is verified.

(vi) `3x^2 – x – 4`

Answer: `3x^2 – x – 4 = 0`
Or, `3x^2 + 3x – 4x – 4 = 0`
Or, `3x(x + 1) – 4(x + 1) = 0`
Or, `(3x – 4)(x + 1) = 0`

Hence, zeroes are; `4/3` and `– 1`

We know that; sum of zeroes `= - b/a`

Or, `4/3-1=1/3`

Or, `(4-3)/(3)=1/3`

We know that; product of zeroes = c/a

Or, `4/3xx(-1)=-4/3`

From sum and product of zeroes, the relationship between the zeroes and coefficients is verified.

Question 2: Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i) ¼ , -1

Answer: We know that a quadratic equation can be given as follows:

`x^2 – text((sum of zeroes))x + text(product of zeroes)`

Hence; the required equation can be written as follows:

`x^2-1/4x-1`

`=4x^2-x-4`

(ii) `sqrt2`, `1/3`

Answer: We know that a quadratic equation can be given as follows:

`x^2 – text((sum of zeroes))x + text(product of zeroes)`

Hence; the required equation can be written as follows:

`x^2-sqrt2x+1/3`

`=3x^2-3sqrt2x+1`

(iii) 0, `sqrt5`

Answer: We know that a quadratic equation can be given as follows:

`x^2 – text((sum of zeroes))x + text(product of zeroes)`

Hence; the required equation can be written as follows:

`x^2-9x+sqrt5`

`=x^2+sqrt5`

(iv) 1, 1

Answer: We know that a quadratic equation can be given as follows:

`x^2 – text((sum of zeroes))x + text(product of zeroes)`

Hence; the required equation can be written as follows:

`x^2 – x + 1`

(v) – ¼, ¼

Answer: We know that a quadratic equation can be given as follows:

`x^2 – text((sum of zeroes))x + text(product of zeroes)`

Hence; the required equation can be written as follows:

`x^2+1/4x+1/4`

`=4x^2+x+1`

(vi) 4, 1

Answer: We know that a quadratic equation can be given as follows:

`x^2 – text((sum of zeroes))x + text(product of zeroes)`

Hence; the required equation can be written as follows:

`x^2 – 4x + 1`