If α and β are the zeroes of the polynomial x2−3x−2, find the quadratic polynomial whose zeroes are 1/2α+β and 1/2β+α.
By BYJU'S Exam Prep
Updated on: October 17th, 2023
If α and β are the zeroes of the polynomial x2−3x−2, the quadratic polynomial whose zeroes are 1/2α+β and 1/2β+α is 16x2−9x+16. Before attempting to answer a polynomial equation, candidates should express it in standard form. Factor it, then when each variable factor reaches zero, set them all to zero. The answers to the derived equations are the solutions to the original equations.
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x2−3x−2, Find a polynomial whose zeros are 1/2α+β and 1/2β+α
The question states If α and β are the zeroes of the polynomial x2−3x−2, find the quadratic polynomial whose zeroes are 1/2α+β and 1/2β+α.” A polynomial function can only have real roots that are at most equal to its degree. Find a function’s roots by setting the value of the function to zero. Following are the methods for finding a polynomial whose zeros are 1/2α+β and 1/2β+α:
Given that α and β are zero of polynomial
f(x) = x2−3x−2
α+β = 3
αβ = −2
Now, the zero of the required quadratic polynomial are 1/2α+β and 1/2β+α
Sum of zeros = 1/2α+β + 1/2β+α
(2α+β)(2β+α)/2β+α+2α+β
3(α+β)/4αβ+2α2+2β2+αβ
9/ (-10+26)
9/ 16
Product of zeros = 1/2α+β * 1/2β+α
1/ 4αβ + 2[(α+β)2 – 2αβ] + αβ
1/ 16
Required equation,
x2−(sum of roots)x+ Product of roots=0
16×2−9x+16 = 0
Hence, the quadratic polynomial whose zeroes are 1/2α+β and 1/2β+α is 16×2−9x+16.
Summary:
If α and β are the zeroes of the polynomial x2−3x−2, find the quadratic polynomial whose zeroes are 1/2α+β and 1/2β+α.
The quadratic polynomial whose zeroes are 1/2α+β and 1/2β+α is 16×2−9x+16 if α and β are the zeroes of the polynomial x2−3x−2. As everyone is aware, the product of zeros equals c/a and the sum of zeros equals b/a. A quadratic polynomial is represented by the expression k [x2 – (zero sum)x + (zero product)]. We can obtain the desired polynomial, whose zeros are 1/2α+β and 1/2β+α, by substituting all values.
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