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AnilSinghBora
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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, 1/4vi) 4, 1 Q.2

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Please guide me for solving this question. The polynomials chapter is very important. I want the solution of this exercise 2.2 question no.2 solution. How can i easily solve it. 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, 1/4vi) 4, 1

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  1. (i) 1/4 , -1

    Solution:

    From the formulas of sum and product of zeroes, we know,

    Sum of zeroes = α+β

    Product of zeroes = α β

    Sum of zeroes = α+β = 1/4

    Product of zeroes = α β = -1

    ∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

    x2–(α+β)x +αβ = 0

    x2–(1/4)x +(-1) = 0

    4x2–x-4 = 0

    Thus,4x2–x–4 is the quadratic polynomial.

    (ii)√2, 1/3

    Solution:

    Sum of zeroes = α + β =√2

    Product of zeroes = α β = 1/3

    ∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

    x2–(α+β)x +αβ = 0

    x2 –(√2)x + (1/3) = 0

    3x2-3√2x+1 = 0

    Thus, 3x2-3√2x+1 is the quadratic polynomial.

    (iii) 0, √5

    Solution:

    Given,

    Sum of zeroes = α+β = 0

    Product of zeroes = α β = √5

    ∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly

    as:-

    x2–(α+β)x +αβ = 0

    x2–(0)x +√5= 0

    Thus, x2+√5 is the quadratic polynomial.

    (iv) 1, 1

    Solution:

    Given,

    Sum of zeroes = α+β = 1

    Product of zeroes = α β = 1

    ∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

    x2–(α+β)x +αβ = 0

    x2–x+1 = 0

    Thus , x2–x+1is the quadratic polynomial.

    (v) -1/4, 1/4

    Solution:

    Given,

    Sum of zeroes = α+β = -1/4

    Product of zeroes = α β = 1/4

    ∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

    x2–(α+β)x +αβ = 0

    x2–(-1/4)x +(1/4) = 0

    4x2+x+1 = 0

    Thus,4x2+x+1 is the quadratic polynomial.

    (vi) 4, 1

    Solution:

    Given,

    Sum of zeroes = α+β =

    Product of zeroes = αβ = 1

    ∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

    x2–(α+β)x+αβ = 0

    x2–4x+1 = 0

    Thus, x2–4x+1 is the quadratic polynomial.

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