Question

If alpha and beta are the zeroes of the polynomial f (x) =3x2+5x+7 then find the value of 1/alpha2+1/beta

Answer 1

$\huge{\underline{\bf{\blue{Question:-}}}}$

If alpha and beta are the zeroes of the polynomial f (x) =3x2+5x+7 then find the value of 1/alpha2+1/beta

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$\large{\underline{\bf{\pink{Answer:-}}}}$

The value of 1/α + 1/β = -5/7

$\large{\underline{\bf{\purple{Explanation:-}}}}$

$\large{\underline{\bf{\green{Given:-}}}}$

⠀⠀⠀⠀⠀⠀⠀f(x) = 3x² + 5x + 7

$\large{\underline{\bf{\green{To\:Find:-}}}}$

we need to find the Value of 1/α + 1/β.

$\huge{\underline{\bf{\red{Solution:-}}}}$

If α and β are zeroes of polynomial 3x² + 5x + 7.

We know that,

If α and β are the zeroes quadratic polynomial of ax² + bx + c .

Then,

$\boxed{\bf{sum \: of \: zeroes \alpha + \beta = \frac{ - b}{a} }} \\ \\ \boxed{\bf{product \: of \: zeroes \alpha \beta = \frac{c}{a}} }$

✰ ⠀⠀⠀⠀⠀α + β = -5/3

✰ ⠀⠀⠀⠀⠀α β = 7/3

Now,

$\bf :\implies \frac{1}{ \alpha } + \frac{1}{ \beta } \\ \\ \bf \: :\implies \frac{ \beta \alpha }{ \alpha \beta }$

$: \implies\bf \: \frac{ \frac{ \frac{ - 5}{3} }{7} }{3} \\ \\ : \implies\bf \: \frac{ - 5}{3} \times \frac{3}{7} \\ \\: \implies\bf \: \frac{ - 5}{7}$

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Answer 2

Given:

f(x)=3x²+ 5x + 7

To Find:

$\sf{ \frac{1}{ \alpha } + \frac{1 }{ \beta }}$

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Solution:

$\sf{ \alpha \: and \: \beta }$ are the zeros of the polynomial 3x² +5x + 7.

Therefore, $\sf{ \alpha \: and \: \beta}$ are the zeros of the quadratic polynomial ax² + bx + c

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Sum of the zeros:

$\sf{\implies \alpha + \beta = \frac{ - b}{a}}$

$\sf{\implies \alpha + \beta = \frac{ - 5}{3}}$

Product of the zeros:

$\sf{\implies \alpha \beta = \frac{c}{a}}$

$\sf{\implies \alpha \beta = \frac{7}{3}}$

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Now,

$\sf{\implies \frac{1}{ \alpha } + \frac{1 }{ \beta }}$

$\sf{\implies \frac{ \beta + \alpha }{ \alpha \beta } }$

$\sf{\implies \frac{ \frac{ -5 }{3} }{ \frac{7}{3} } }$

$\sf{\implies \frac{ - 5}{3} \times \frac{3}{7} }$

$\sf{\implies \frac{ - 5}{7} }$

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$\huge\underline\mathfrak\red{ThankYou}$