Find the value of k, if (x+3) is a factor of 3x²+kx+6 ​

Question

Find the value of k, if (x+3) is a factor of 3x²+kx+6

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Arianna 3 months 2021-10-15T17:17:56+00:00 2 Answers 0 views 0

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    0
    2021-10-15T17:18:59+00:00

     \sf \red{Poynomial \implies \: x + 3} \\  \sf \green{Factor\: of \: poynomial \implies \:  {3x}^{2}  + kx + 6}

    \rule{200}2

     \implies \sf{x + 3 = 0} \\  \implies \sf{x = 0 - 3}  \\  \therefore \sf{x =  - 3}

    \rule{200}2

     \implies \sf \purple{ {3x}^{2}  + kx + 6 = 0} \\  \implies \sf{ 3 \times ( - 3)  \times ( - 3) + k( - 3) + 6 = 0} \\  \implies \sf{27 + ( - 3)k + 6 = 0} \\  \implies \sf{27 - 3k + 6 = 0}  \\  \implies \sf{33 - 3k = 0} \\  \implies \sf{ - 3k = 0 - 33}  \\  \implies \sf{ - 3k =  - 33} \\  \implies \sf{k =  \frac{33}{3} } \\    \sf \therefore  \orange{k = 11}

    \rule{200}2

     \star \:  \bf \underline  \pink{Verification : }

     \implies \sf \purple{ {3x}^{2}  + kx + 6 } \\  \implies \sf{ 3 \times ( - 3)  \times ( - 3) + 11( - 3) + 6 } \\  \implies \sf{27 + ( - 3)11 + 6 } \\  \implies \sf{27 - 3k\times\:11 + 6 }  \\  \implies \sf{33 - 3(11) } \\  \implies \sf{ 33-33}  \\   \sf \therefore  \orange{0}

    \bf\green{Hence,verified}

    0
    2021-10-15T17:19:29+00:00

    Given:

    Polynomial = P(x) = 3x² + kx + 6

    Factor of the ablove polynomial = (x+3)

    To be found:

    Value of K, for which (x+3) become the factor of P(x) = 3x² + kx + 6

    Now,

    x + 3 = 0

    ⇒ x = (-3)

    So,

    As (x+3) is a factor so x = (-3) is one root of the polynomial.

    Therefore,

    P(-3) = 0

    → P(-3) = 3(-3)² + k(-3) + 6 = 0

    → 3(9) – 3k + 6 = 0

    → 27 – 3k + 6 = 0

    → 27 + 6 – 3k = 0

    → 33 – 3k = 0

    → – 3k = -33

    → k = -33 ÷ -3

    → k = 11

    Hence,

    For the value of k = 11, (x+3) is a factor of 3x²+ kx + 6

    Verification:

    3x²+ kx + 6, by putting the value of k = 11 and taking -3 as root the remainder should be zero

    = 3x²+ 11x + 6

    = 3(-3)² + 11(-3) + 6

    = 3(9) – 33 + 6

    = 27 – 33 + 6

    = 27 + 6 – 33

    = 33 – 33

    = 0

    Hence verified.

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