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`