Answer:
see explanation
Step-by-step explanation:
Given
p(x) = 2x² - 3x + 7
with a = 2, b = - 3, c = 7
If α and β are the zeros, then
the sum of the roots
α + β = - [tex]\frac{-3}{2}[/tex] = [tex]\frac{3}{2}[/tex] ←
and the product of the roots
αβ = [tex]\frac{c}{a}[/tex] = [tex]\frac{7}{2}[/tex] ←
(a)
α² + β² = (α + β)² - 2αβ = ([tex]\frac{3}{2}[/tex] )² - 2 × [tex]\frac{7}{2}[/tex] = [tex]\frac{9}{4}[/tex] - 7 = - [tex]\frac{19}{4}[/tex]
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(b)
α³ + β³
= (α + β)³ - 3α²β - 3αβ²
= (α + β)³ - 3α(α + β)
= ([tex]\frac{3}{2}[/tex] )³ - 3 × [tex]\frac{7}{2}[/tex] × [tex]\frac{3}{2}[/tex]
= [tex]\frac{27}{8}[/tex] - [tex]\frac{63}{4}[/tex] = - [tex]\frac{99}{8}[/tex]
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(c)
[tex]\frac{1}{\alpha }[/tex] + [tex]\frac{1}{\beta }[/tex]
= [tex]\frac{\alpha+\beta }{\alpha\beta }[/tex]
= [tex]\frac{3}{2}[/tex] × [tex]\frac{2}{7}[/tex] = [tex]\frac{3}{7}[/tex]
