SOLUTION: if f(x)=2x show that f (x + 4)-f(x-2)=63/4 if f(x,y) = x4 + 3x2 y2 + y4 show that f (ax,ay)= a4.f(x,y)

Algebra ->  Proofs -> SOLUTION: if f(x)=2x show that f (x + 4)-f(x-2)=63/4 if f(x,y) = x4 + 3x2 y2 + y4 show that f (ax,ay)= a4.f(x,y)      Log On


   

Question 369957: if f(x)=2x
show that f (x + 4)-f(x-2)=63/4
if f(x,y) = x4 + 3x2 y2 + y4
show that f (ax,ay)= a4.f(x,y)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%2Cy%29+=+x%5E4+%2B+3x%5E2y%5E2+%2B+y%5E4 Start with the given equation.


f%28ax%2Cay%29+=+%28ax%29%5E4+%2B+3%28ax%29%5E2%28ay%29%5E2+%2B+%28ay%29%5E4 Replace each 'x' with 'ax'. Replace each 'y' with 'ay'


f%28ax%2Cay%29+=+a%5E4x%5E4+%2B+3a%5E2x%5E2a%5E2y%5E2+%2B+a%5E4y%5E4 Use the property %28xy%29%5Ez=x%5Ezy%5Ez


f%28ax%2Cay%29+=+a%5E4x%5E4+%2B+3a%5E2%2Aa%5E2x%5E2y%5E2+%2B+a%5E4y%5E4 Rearrange the terms.


f%28ax%2Cay%29+=+a%5E4x%5E4+%2B+3a%5E4x%5E2y%5E2+%2B+a%5E4y%5E4 Multiply a%5E2 and a%5E2 to get a%5E2%2Aa%5E2=a%5E%282%2B2%29=a%5E4


f%28ax%2Cay%29+=+a%5E4%28x%5E4+%2B+3x%5E2y%5E2+%2B+y%5E4%29 Factor out the GCF a%5E4


f%28ax%2Cay%29+=+a%5E4f%28x%2Cy%29 Now replace the expression in the parenthesis with f%28x%2Cy%29 (this is possible since f%28x%2Cy%29+=+x%5E4+%2B+3x%5E2y%5E2+%2B+y%5E4)