If α and β are the zeroes of the polynomial 2x² - x – 3 , then write the polynomial whose
zeroes are α², β².
Answers
Answer:
2x² - 5x + 82 is the required polynomial
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Explainaion:
α² + β² can be written as (α + β)² - 2αβ
p(x) = 2x² - 5x + 7
a = 2 , b = - 5 , c = 7
α and β are the zeros of p(x)
we know that ,
sum of zeros = α + β
= -b/a
= 5/2
product of zeros = c/a
= 7/2
Now
2α + 3β and 3α + 2β are zeros of a polynomial.
sum of zeros = 2α + 3β+ 3α + 2β
= 5α + 5β
= 5 [ α + β]
= 5 × 5/2
= 25/2
product of zeros = (2α + 3β)(3α + 2β)
= 2α [ 3α + 2β] + 3β [3α + 2β]
= 6α² + 4αβ + 9αβ + 6β²
= 6α² + 13αβ + 6β²
= 6 [ α² + β² ] + 13αβ
= 6 [ (α + β)² - 2αβ ] + 13αβ
= 6 [ ( 5/2)² - 2 × 7/2 ] + 13× 7/2
= 6 [ 25/4 - 7 ] + 91/2
= 6 [ 25/4 - 28/4 ] + 91/2
= 6 [ -3/4 ] + 91/2
= -18/4 + 91/2
= -9/2 + 91/2
= 82/2
= 41
-18/4 = -9/2 [ simplest form ]
a quadratic polynomial is given by :-
k { x² - (sum of zeros)x + (product of zeros) }
k {x² - 5/2x + 41}
k = 2
2 {x² - 5/2x + 41 ]
2x² - 5x + 82 is the required polynomial
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Answer:
Step-by-step explanation:
Given that α and β are roots of 2x² - x - 3
2x² - x - 3
=> 2x² - 3x + 2x - 3
=> x(2x - 3) + 1(2x - 3)
=> (x + 1)(2x - 3)
Hence either α = (x + 1) , β = (2x - 3)
or β = (x + 1) , α = (2x - 3)
So we have to find a polynomial whose roots are α²,β²
which are (x + 1)²,(2x - 3)²
That is simply the multiplication of (x + 1)²(2x - 3)²
=> (x² + 2x + 1)(4x² - 12x + 9)
=>
=>
So this is the answer.
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