TOPICS

Exercise - 2.2

Polynomials

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

(i) x² – 2x – 8

(ii) 4s² – 4s + 1

(iii) 6x² – 3 – 7x

(iv) 4u² + 8u

(v) t² – 15

(vi) 3x² – x – 4

(i) p(x) = x² – 2x – 8 = x² – 4x + 2x – 8 = x(x - 4) + 2(x - 4) = (x + 2)(x - 4) Zeroes of polynomial : x + 2 = 0 x = -2 x - 4 = 0 x = 4 Relationship between the zeroes and the coefficients : p(x) = x² – 2x – 8 a = 1, b = -2, c = -8 Sum of zeroes = α + β = -b/a = -(-2)/1 = 2 Product of zeroes = αβ = c/a = -8/1 = -8

(ii) p(s) = 4s² – 4s + 1 = 4s² – 2s - 2s + 1 = 2s(2s - 1) - 1(2s - 1) = (2s - 1)(2s - 1) Zeroes of polynomial : 2s - 1 = 0 2s = 1 s = 1/2, 1/2 Relationship between the zeroes and the coefficients : p(s) = 4s² – 4s + 1 a = 4, b = -4, c = 1 Sum of zeroes = α + β = -b/a = -(-4)/4 = 1 Product of zeroes = αβ = c/a = 1/4

(iii) p(x) = 6x² – 3 – 7x = 6x² – 7x - 3 = 6x² – 9x + 2x – 3 = 3x(2x - 3) + 1(2x - 3) = (2x - 3)(3x + 1) Zeroes of polynomial : 2x - 3 = 0 2x = 3 x = 3/2 3x + 1 = 0 3x = -1 x = -1/3 Relationship between the zeroes and the coefficients : p(x) = 6x² – 3 – 7x a = 6, b = -7, c = -3 Sum of zeroes = α + β = -b/a = -(-7)/6 = 7/6 Product of zeroes = αβ = c/a = -3/6 = -1/2

(iv) p(u) = 4u² + 8u = 4u(u + 2) Zeroes of polynomial : 4u = 0 u = 0 u + 2 = 0 u = -2 Relationship between the zeroes and the coefficients : p(u) = 4u² + 8u a = 4, b = 8, c = 0 Sum of zeroes = α + β = -b/a = -(8)/4 = -2 Product of zeroes = αβ = c/a = 0/4 = 0

(v) p(t) = t² – 15 = t² - (√15)² = (t + √15)(t - √15) Zeroes of polynomial : t + √15 = 0 t = -√15 t - √15 = 0 t = √15 Relationship between the zeroes and the coefficients : p(t) = t² – 15 a = 1, b = 0, c = -15 Sum of zeroes = α + β = -b/a = 0/1 = 0 Product of zeroes = αβ = c/a = -15/1 = -15

(vi) p(x) = 3x² – x – 4 = 3x² – 4x + 3x – 4 = x(3x - 4) + 1(3x - 4) = (3x - 4)(x + 1) Zeroes of polynomial : 3x - 4 = 0 3x = 4 x = 4/3 x + 1 = 0 x = -1 Relationship between the zeroes and the coefficients : p(x) = 3x² – x – 4 a = 3, b = -1, c = -4 Sum of zeroes = α + β = -b/a = -(-1)/3 = 1/3 Product of zeroes = αβ = c/a = -4/3

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

(i) 1/4, -1 (ii) √2, 1/3 (iii) 0, √5 (iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1.

(i) 1/4, -1 Given that : Sum of zeroes = α + β = -b/a = 1/4 Product of zeroes = αβ = c/a = -4/4 Therefore, a = 4, b = -1, c = -4. In general form, p(x) = ax² + bx + c. So, quadratic polynomial is 4x² - x - 4.

(ii) √2, 1/3 Given that : Sum of zeroes = α + β = -b/a = 3√2/3 Product of zeroes = αβ = c/a = 1/3 Therefore, a = 3, b = -3√2, c = 1. In general form, p(x) = ax² + bx + c. So, quadratic polynomial is 3x² - 3√2x + 1.

(iii) 0, √5 Given that : Sum of zeroes = α + β = -b/a = 0/1 Product of zeroes = αβ = c/a = √5/1 Therefore, a = 1, b = 0, c = √5. In general form, p(x) = ax² + bx + c. So, quadratic polynomial is x² + √5.

(iv) 1, 1 Given that : Sum of zeroes = α + β = -b/a = 1/1 Product of zeroes = αβ = c/a = 1/1 Therefore, a = 1, b = -1, c = 1. In general form, p(x) = ax² + bx + c. So, quadratic polynomial is x² - x + 1.

(v) -1/4, 1/4 Given that : Sum of zeroes = α + β = -b/a = -1/4 Product of zeroes = αβ = c/a = 1/4 Therefore, a = 4, b = 1, c = 1. In general form, p(x) = ax² + bx + c. So, quadratic polynomial is 4x² + x + 1.

(vi) 4, 1 Given that : Sum of zeroes = α + β = -b/a = 4/1 Product of zeroes = αβ = c/a = 1/1 Therefore, a = 1, b = -4, c = 1. In general form, p(x) = ax² + bx + c. So, quadratic polynomial is x² - 4x + 1.

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