if alpha [tex] \alpha and \beta are the zeroes \: of \: the \: polynomial \: 2 x { + 5x + 1 \: }^{2} then \: the \: value \: of \alpha

Question

if alpha
 \alpha and \beta are  the zeroes \: of \: the \: polynomial \: 2 x { + 5x + 1 \: }^{2} then \: the \: value \: of \alpha  +  \beta  +  \alpha  \times  \beta

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Katherine 1 month 2021-10-28T00:11:12+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-10-28T00:12:15+00:00

    \large{\red{\underline{\underline{\bold{Given:-}}}}}

    • A quadratic polynomial 2x² + 5x + 1

    \sf \alpha  \: and \:  \beta  \: are \: the \: zeroes \: of \: the \: given \: quadratic \: polynomial

    \large{\blue{\underline{\underline{\bold{To\:Find:-}}}}}

     \sf the \: value \: of \: \alpha   + \beta   +  \alpha    \beta

    \large{\green{\underline{\underline{\bold{Solution:-}}}}}

    For a quadratic polynomial ax² + bx + c, if the zeroes are α and β

    \sf Sum \: of \: zeroes( \: \alpha   + \beta )   =  \dfrac{ - b}{a}

    \sf Product \: of \: zeroes( \: \alpha  \beta )   =  \dfrac{ c}{a}

    Compare 2x² + 5x + 1 with ax² + bx + c

    • a = 2
    • b = 5
    • c = 1

    \sf sum \: of \: zeroes( \: \alpha   + \beta )   =  \dfrac{ - b}{a}   =  \dfrac{ - 5}{2}

    \sf product \: of \: zeroes( \: \alpha   \beta )   =  \dfrac{ c}{a}   =  \dfrac{1 }{2}

    Now:-

    \sf  \implies \alpha  +  \beta  +  \alpha  \beta

    \sf  \implies  \dfrac{ - 5}{2}  +  \dfrac{1}{2}

    \sf  \implies  \dfrac{ - 5 + 1}{2}

    \sf  \implies  \dfrac{ - 4}{2}

    \sf  \implies  -2

    Hence;

     \large \boxed{ \sf  \purple{ \implies \alpha  +  \beta  +  \alpha  \beta =     - 2}}

    0
    2021-10-28T00:12:52+00:00

    Answer

    α + β + αβ = – 2

    Explanation

    Compare given equation 2x² + 5x + 1 with ax² + bx + c , we get ,

    • a = 2 , b = 5 , c = 1

    also given α and β are zeroes of polynomial .

    Let ,

    Sum of zeroes , α + β = -b/a = -(5)/2

    α + β = – ⁵/₂ … (1)

    Product of zeroes , αβ = c/a = (1)/(2)

    αβ = ¹/₂ … (2)

    Now our required ,

    α + β + αβ

    ⇒ ( α + β ) + ( αβ )

    ⇒ ( – ⁵/₂ ) + ( ¹/₂ ) [ From (1) & (2) ]

    ⇒ ( ⁻ ⁵ ⁺ ¹/₂ )

    ⇒ ⁻⁴/₂

    – 2

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    https://brainly.in/question/19923032

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