Find the remainder when x^2020 + x^1947 + 1 is divided by x^2 – 1

Question

Find the remainder when x^2020 + x^1947 + 1 is divided by x^2 – 1

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Brielle 5 months 2022-01-06T01:22:23+00:00 1 Answer 0 views 0

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    2022-01-06T01:23:29+00:00

    Answer:

    x + 2

    Step-by-step explanation:

    The degree of the remainder is at most 1 since we’re dividing by a degree 2 polynomial.  So the remainder takes the form ax+b.  We just need to work out a and b.

    When we do the division, since ax+b is the remainder, we get

    x^2020 + x^1947 + 1 = ( x² – 1 ) g(x) + ax + b.

    Since x² – 1 vanishes when we put x = 1 and x = -1, we put these values in:

    Putting x = 1 gives 3 = a + b.  (*)

    Putting x = -1 gives 1 = -a + b.  (**)

    Subtracting (**) from (*) gives 2a = 2, so a = 1.

    Adding (*) and (**) gives 2b = 4, so b = 2.

    Therefore the remainder is ax+b = x+2.

    Hope this helps.

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