Question

if aplha,beta are the zeros of the polynomial p(x) = x^2+x+1,then 1/alpha + 1/beta is​

Answer 1

$\large \bf \clubs \:  Given :-$

• α and β are the zeros polymomial

p(x) = x² + x + 1.

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$\large \bf \clubs \:  To \: Find :-$

Value of $\sf\dfrac{1}{\alpha}+\dfrac{1}{\beta}$

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$\large \bf \clubs \:  Main \: Formula :-$

☆ Relationship between Zeros and coefficiants of a Quadratic Polynomial :

For a qudratic polynomial of the Form ax² + bx + c

• Sum of Zeros =$\sf-\dfrac{b}{a}$

• Product of Zeros =$\sf\dfrac{c}{a}$

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$\large \bf \clubs \:  Solution :-$

We Have ,

α and β are the zeros polymomial

p(x) = x² + x + 1.

Hence ,

$\large \pmb{ \alpha + \beta = - 1 } \: \: \:----(1) \: \\ \\ \large\pmb{ \alpha \beta = 1} \: \: \:---- (2)$

Now,

$\large \dfrac{1}{ \alpha } + \frac{1}{ \beta } \\ \\ \large \sf = \frac{ \beta + \alpha }{ \alpha \beta } \\ \\ \bf \: \: \: \: \{using \: (1) \: and \: (2) \} \\ \\ \large\sf = \frac{ - 1}{1} \\ \\ = -1 \\ \\ \purple{ \Large :\longmapsto  \underline { \pmb{\boxed{{  \frac{1}{ \alpha } + \frac{1}{ \beta } = - 1 } }}}}$

$\Large\red{\mathfrak{  \text{W}hich \:\:is\:\: the\:\: required} }\\ \LARGE \red{\mathfrak{ \text{ A}nswer.}}$

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Answer 2

Answer:

Given :

• p(x) = x² + x + 1
• α and β are zeros of p(x)

Solution

x² + x + 1

Here :-

• a = 1
• b = 1
• c = 1

Sum of zeroes : α + β = (-b/a)

• α + β = ( -1/1)
• α + β = ( -1 )

Product of zeroes : α x β = c/a

• α x β = (1/1)
• α x β = 1

To find :

• 1/α + 1/β

Answer :

• 1/α + 1/β
• ( β + α ) / αβ
• (-1)/1
• (-1)

Answer = (-1)

Hope it helps you :)