Question

Find the remainder when we divide \[{x^7}y - x{y^7}\;\] by \[(x + y)({x^2} - xy + {y^2})\].

Answer 1

Hint: We will use the theorem of remainder here to deal with this problem. As we are dividing by \[(x + y)({x^2} - xy + {y^2})\], so it should be divisible by \[(x + y)\]also, hence substituting \[x = - y\] in the equation of remainder theorem and on simplification we will get our needed remainder.

Complete step-by-step answer:
We have to divide, \[{x^7}y - x{y^7}\;\]by \[(x + y)({x^2} - xy + {y^2})\] and find the needed remainder.
Now, as per the remainder theorem, we get,
 \[{x^7}y - x{y^7}\; = (x + y)({x^2} - xy + {y^2})Q + R\] ……………(1)
Where Q is the dividend and R is our needed remainder.
Now, as if it is divisible by \[(x + y)({x^2} - xy + {y^2})\] it should be divisible by \[(x + y)\] also.
Then putting \[x = - y\] will give us our desired remainder.
So, substituting, \[x = - y\] in (1) we get,
 \[{( - y)^7}.y - ( - y).{y^7}\; = ( - y + y)({( - y)^2} - ( - y).y + {y^2})Q + R\]
On simplification we get,
 \[ \Rightarrow - {y^8} + {y^8} = 0.Q + R\]
Hence on solving for R we get,
 \[ \Rightarrow R = 0\]
So, we have the remainder as, 0.

Note: The Remainder Theorem starts with an unnamed polynomial \[p\left( x \right)\], where " \[p\left( x \right)\]" just means "some polynomial p whose variable is . Then the Theorem talks about dividing that polynomial by some linear factor \[x\;-\;a\], where a is just some number. Then, as a result of the long polynomial division, you end up with some polynomial answer q(x) (the "q" standing for "the quotient polynomial") and some polynomial remainder \[r\left( x \right)\].