POLYNOMIALS

posted by Aoi

I'm really sorry, i'm preparing for O levels, so i have a lot of maths questions to ask... please excuse me...

Question:

Find the value of k for which a-3b is a factor of a^4 - 7a^2b^2 + kb^4. Hence, factorise completely.

How do i even find k???


Please help thankyou!

  1. Reiny

    Using either long division or by synthetic division I go

    (a^4 - 7a^2b^2 + kb^4)÷ (a-3b) = a^3 + 3a^2b - 7ab^2 + 21b^3 with a remainder of 63k^4 + kb^4

    but there was to be no remainder, so
    63k^4 + kb^4 = 0
    k = -63

    so a^3 + 3a^2b - 7ab^2 + 21b^3 = 0
    use grouping
    a^2(a+3b) + 7b^2(a+3b) = 0
    (a+3b)(a^2 + 7b) = 0

    so the original factors to
    (a-3b)(a+3b)(a^2 + 7b)

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