Question

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!

Answer 1

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)