Question

Find the ratio of the sum of the zeroes to product of the zeroes of the cubic polynomial
ax³ + bx² + cx +d

Answer 1

$Given \: Cubic \: polynomial :\\ax^{3} + bx^{2} + cx + d$

$i )Sum \:of \:the \: zeroes = \frac{ - x^{2} - coefficient }{x^{3} - coefficient }\\= \frac{-b}{a} \:--(1)$

$ii ) Product\:of \:the \: zeroes = \frac{ - (constant\:term) }{x^{3} - coefficient }\\= \frac{-d}{a} \:--(2)$

$Ratio \:of \: the \: sum \:of \:the \: zeroes \:to \\product \:of \:the \: zeroes \:of \:the \\Cubic \: Polynomial = \frac{ \frac{-b}{a}}{\frac{-d}{a}}\\= \frac{b}{d} \\= b : d$

Therefore.,

$\red { Required \:ratio } \green { = b : d }$

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Answer 2

$\mathtt{ \huge{ \fbox{Solution :)}}}$

We know that ,

The sum of roots of cubic polynomial is given by

$\large \mathtt{ \fbox{α + β + γ = - \frac{b}{a} }}$

and

The product of roots of cubic polynomial is given by

$\large \mathtt{ \fbox{α \times β \times γ = - \frac{d}{a} }}$

Thus , the ratio of sum and product of roots of cubic polynomial is

$\sf \hookrightarrow \frac{ \alpha + β + γ}{ \alpha \times β \times γ} = \frac{ \: \: - \frac{ b}{a} \: \: }{ - \frac{d}{a} } \\ \\ \sf \hookrightarrow \frac{ \alpha + β + γ}{ \alpha \times β \times γ} = \frac{b}{d}$

Hence , the ratio is b : d

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