If polynomial x4 − 2ax3 + 5×2– 3x + 1 is divided by x + 2, it leaves 3 as remainder, then find the value of ‘a’​

Question

If polynomial x4 − 2ax3

+ 5×2– 3x + 1 is divided by x + 2, it leaves 3 as remainder, then find the value of ‘a’​

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Cora 1 month 2021-09-22T21:27:29+00:00 1 Answer 0 views 0

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    2021-09-22T21:29:14+00:00

     

    There is theorem known as “Polynomial Remainder Theorem” or “ Bezout’s Theorem”. It is Stated as –

    A Polynomial f(x) if divided by a linear polynomial (x-a) leaves remainder which equals f(a).

    So , getting back to our question –

    f(x) = x^4 – 2x^3 + 3x^2 – ax + b

    So , when it is divided by (x – 1) it’ll leave a remainder = f(1) = 5 (Given).

    f(1) = 1^4 – 2×1^3 + 3×1^2 – a×1 + b = 5

    => 1 – 2 + 3 – a + b = 5

    => a – b = (-3) …. Eqn(1)

    Now , Similarly –

    f(-1) = (-1)^4 – 2×(-1)^3 + 3×(-1)^2 – a×(-1) + b = 19

    => 1 + 2 + 3 + a + b = 19

    => a + b = 13 …. Eqn(2)

    Now , adding equations (1) and (2) , We’ll get –

    (a+b) + (a-b) = (-3) + 13

    => 2a = 10 => a = 5

    So , (a +b) = 13 implies b = 8

    Hence , Values of a and b are 5 and 8 respectively.

    Done!

    HOPE THIS HELPS…☺☺☺

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