Question

If alpha beta gamma delta are the roots of the equation x⁴ - 4x³ + 5x² - 2x - 2=0, then
product of the roots is​

Answer 1

Answer:

-2

Step-by-step explanation:

the last term (constant) is the product of roots in a polynomial (with the same sign in polynomial of even power) and vice versa.

eg for quadratic product of roots =c/a

Answer 2

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FORMULA TO BE IMPLEMENTED

$\sf{ \alpha \: , \beta , \gamma , \delta \: \: are \: the \: roots \: of \: the \: equation \: \: }$

$a {x}^{4} + b {x}^{3} + c{x}^{2} +d x +e = 0 \: \:$

$\displaystyle \: \sum \alpha = - \frac{b}{a} \:$

$\displaystyle \: \sum \alpha \beta = \frac{c}{a} \:$

$\displaystyle \: \sum \alpha \beta \gamma = - \frac{d}{a} \:$

$\displaystyle \: \alpha \beta \gamma \delta = \frac{e}{a} \:$

GIVEN

$\sf{ \alpha \: , \beta , \gamma , \delta \: \: are \: the \: roots \: of \: the \: equation \: \: }$

${x}^{4} - 4{x}^{3} + 5{x}^{2} - 2 x - 2 = 0 \: \:$

TO DETERMINE

The product of roots

EVALUATION

Comparing the given equation

$\sf{ {x}^{4} - 4{x}^{3} + 5{x}^{2} - 2 x - 2 = 0 \: \: with}$

${a {x}^{4} + b {x}^{3} + c{x}^{2} +d x +e = 0 \: \: we \: get}$

$a = 1 \: , \: b= - 4 \: , \: c= 5 \: , \: d = - 2 \: , \: e = - 2$

RESULT

The required product of the roots

$\displaystyle \: \alpha \beta \gamma \delta = \frac{e}{a} = \frac{ - 2}{1} = - 2$