Question

find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively 3 {x }^{2} - x - 43x 2 −x−4 ​

Answer 1

Answer:

(i) -8/3 , 4/3 A quadratic polynomial formed for the given sum and product of zeros is given by: f(x) = x2 + -(sum of zeros) x + (product of roots) Here, the sum of zeros is = -8/3 and product of zero= 4/3 Thus, The required polynomial f(x) is, ⇒ x2 – (-8/3)x + (4/3) ⇒ x2 + 8/3x + (4/3) So, we put f(x) = 0 ⇒ x2 + 8/3x + (4/3) = 0 ⇒ 3x2 + 8x + 4 = 0 ⇒ 3x2 + 6x + 2x + 4 = 0 ⇒ 3x(x + 2) + 2(x + 2) = 0 ⇒ (x + 2) (3x + 2) = 0 ⇒ (x + 2) = 0 and, or (3x + 2) = 0 Therefore, the two zeros are -2 and -2/3. (ii) 21/8 , 5/16 A quadratic polynomial formed for the given sum and product of zeros is given by: f(x) = x2 - (sum of zeros) x + (product of roots) Here, the sum of zeros is = 21/8 and product of zero = 5/16 Thus, The required polynomial f(x) is, ⇒ x2 – (21/8)x + (5/16) ⇒ x2 – 21/8x + 5/16 So, We put f(x) = 0 ⇒ x2 – 21/8x + 5/16 = 0 ⇒ 16x2 – 42x + 5 = 0 ⇒ 16x2 – 40x – 2x + 5 = 0 ⇒ 8x(2x – 5) – 1(2x – 5) = 0 ⇒ (2x – 5) (8x – 1) = 0 ⇒ (2x – 5) = 0 and, or (8x – 1) = 0 Therefore, the two zeros are 5/2 and 1/8. (iii) -2√3, -9 A quadratic polynomial formed for the given sum and product of zeros is given by: f(x) = x2 - (sum of zeros) x + (product of roots) Here, the sum of zeros is = -2√3 and product of zero = -9 Thus, The required polynomial f(x) is, ⇒ x2 – (-2√3)x + (-9) ⇒ x2 + 2√3x – 9 So, we put f(x) = 0 ⇒ x2 + 2√3x – 9 = 0 ⇒ x2 + 3√3x – √3x – 9 = 0 ⇒ x(x + 3√3) – √3(x + 3√3) = 0 ⇒ (x + 3√3) (x – √3) = 0 ⇒ (x + 3√3) = 0 and, or (x – √3) = 0 Therefore, the two zeros are -3√3 and √3. (iv) -3/2√5, -1/2 A quadratic polynomial formed for the given sum and product of zeros is given by: f(x) = x2 - (sum of zeros) x + (product of roots) Here, the sum of zeros is = -3/2√5 and product of zero = -1/2 Thus, The required polynomial f(x) is, ⇒ x2 – (-3/2√5)x + (-1/2) ⇒ x2 + 3/2√5x – 1/2 So, we put f(x) = 0 ⇒ x2 + 3/2√5x – 1/2 = 0 ⇒ 2√5x2 + 3x – √5 = 0 ⇒ 2√5x2 + 5x – 2x – √5 = 0 ⇒ √5x(2x + √5) – 1(2x + √5) = 0 ⇒ (2x + √5) (√5x – 1) = 0 ⇒ (2x + √5) = 0 and, or (√5x – 1) = 0 Therefore, the two zeros are -√5/2 and 1/√5.Read more on Sarthaks.com - https://www.sarthaks.com/623601/also-find-the-zeros-of-these-polynomials-by-factorization-i-8-3-4-3-ii-21-8-5-16-iii-23-9-iv-25

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