Question

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Answer 1

9th Problem :

If α and β are zeroes of a Quadratic polynomial Z(x) = px² + qx + r then :

★  α and β are roots of Quadratic Equation Z(x) = px² + qx + r = 0

$\mathsf{\bigstar\;\;\; Z(\alpha) = 0}\\\\\\\mathsf{\bigstar\;\;\; Z(\beta) = 0}\\\\\\\mathsf{\bigstar\;\;\; Sum\;of\;the\;Roots\;(\alpha + \beta) = \dfrac{-q}{p}}\\\\\\\mathsf{\bigstar\;\;\; Product\;of\;the\;Roots\;(\alpha.\beta) = \dfrac{r}{p}}$

Given : α and β are the zeroes of quadratic polynomial f(t) = t² - 5t + 3

$\mathsf{\implies Sum\;of\;the\;Roots\;(\alpha + \beta) = \dfrac{5}{1} = 5}$

$\mathsf{\implies Product\;of\;the\;Roots\;(\alpha.\beta) = \dfrac{3}{1} = 3}$

$:\implies$  α⁴β³ + α³β⁴

$:\implies$  α³β³(α + β)

$:\implies$  (αβ)³(α + β)

$:\implies$  (3)³(5)

$:\implies$  (27 × 5)

$:\implies$  135

10th Problem :

Given : α and β are the zeroes of quadratic polynomial f(t) = x² + 7x + 3

$\mathsf{\implies Sum\;of\;the\;Roots\;(\alpha + \beta) = \dfrac{-7}{1} = -7}$

$\mathsf{\implies Product\;of\;the\;Roots\;(\alpha.\beta) = \dfrac{3}{1} = 3}$

$:\implies$  (α - β)²

$:\implies$  α² + β² - 2αβ

$:\implies$  (α + β)² - 2αβ - 2αβ

$:\implies$  (α + β)² - 4αβ

$:\implies$  (-7)² - (4 × 3)

$:\implies$  49 - 12

$:\implies$  37