Express the rational number in the standard form: 132/-428
Answers
Given: p(x) = x² - 3x - 2
Given α and β are the zeroes of quadratic polynomial.
α + β = 3
αβ = -2
(i) 2α + 3β and 3α + 2β
Zeroes of polynomial are (2α + 3β) and (3α + 2β)
Sum(α + β) = 5(α + β) = 15
Product(αβ) = (2α + 3β) * (3α + 2β)
= 6a² + 4αβ + 9αβ + 6β²
= 6(α² + β²) + 13αβ
= 6[(α + β)² - 2αβ] + 13αβ
= 6[(3)² - 2(-2)] + 13(-2)
= 52
Therefore, the polynomial obtained is:
g(x) = x² - (α + β)x + αβ
= x² - 15x + 52
(ii)
Zeroes of polynomials are (α²/β) and (β²/α)
Sum(α + β) = α²/β + β²/α
= (α³ + β³)/αβ
= [(α + β)(α² + β² + 2αβ]/αβ
= [(3)(13 - 4)]/-2
= -27/2
Product(αβ) = (α²/β) * (β²/α)
= αβ
= -2
Therefore, the polynomial obtained is:
g(x) = x² - (α + β) + αβ
= x² - (-27/2) + (-2)
= x² + 27/2 - 2
Step-by-step explanation:
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