Asked by helloooo :)
Determine the number of factors for the polynomials x^1000000 - y^1000000
What exactly does it mean to find the number of factors? and how would you do that?
What exactly does it mean to find the number of factors? and how would you do that?
Answers
Answered by
helloooo :)
ps This was the answer my teacher gave. I just don't understand exactly why or how she got it though.
(x^500000 + y^500000)(x^500000 - y^500000)
(x^500000 + y^500000)(x^250000 + y^250000)(x^250000 - y^250000)
...
(x^500000 + y^500000)(x^250000 + y^250000)(x^125000 + y125000)
this is the part I don't know how she got to:
(x^62500 + y^62500)(x^31250 + y^31250)(x^15625 + y^15625)(x^15625 - y^15625)
⬆️
7 factors
(x^500000 + y^500000)(x^500000 - y^500000)
(x^500000 + y^500000)(x^250000 + y^250000)(x^250000 - y^250000)
...
(x^500000 + y^500000)(x^250000 + y^250000)(x^125000 + y125000)
this is the part I don't know how she got to:
(x^62500 + y^62500)(x^31250 + y^31250)(x^15625 + y^15625)(x^15625 - y^15625)
⬆️
7 factors
Answered by
oobleck
a^2 - b^2 = (a-b)(a+b)
as long as a and b are also squares, you can repeat the process.
Note that in the last step, the exponents are no longer even, so the terms ar no longer perfect squares.
as long as a and b are also squares, you can repeat the process.
Note that in the last step, the exponents are no longer even, so the terms ar no longer perfect squares.
Answered by
Bosnian
a^2 - b^2 = ( a + b ) ∙ ( a - b )
x^1000000 - y^1000000 = (x^500000)^2 - (y^500000)^2 =
( x^500000 + y^500000 ) ∙ ( x^500000 - y^500000 )
x^500000 - y^500000 = ( x^250000 )^2 - ( y^250000 )^2 =
( x^250000 + y^250000 ) ∙ ( x^250000 - y^250000 )
x^250000 - y^250000 = ( x^125000 )^2 - ( y^125000 )^2 =
( x^125000 + y^125000 ) ∙ ( x^125000 - y^125000 )
x^125000 - y^125000 = ( x^62500 )^2 - ( y^62500 )^2 =
( x^62500 + y^62500 ) ∙ ( x^62500 - y^62500 )
x^62500 - y^62500 = ( x^31250 )^2 - ( y^31250 )^2 =
( x^31250 + y^31250 ) ∙ ( x^31250 - y^31250 )
x^31250 - y^31250 = ( x^15625 )^2 - ( y^15625 )^2 =
( x^15625 + y^15625 ) ∙ ( x^15625 - y^15625 )
15625 can be further factorized because 15625 = 5^6
x^1000000 - y^1000000 = (x^500000)^2 - (y^500000)^2 =
( x^500000 + y^500000 ) ∙ ( x^500000 - y^500000 )
x^500000 - y^500000 = ( x^250000 )^2 - ( y^250000 )^2 =
( x^250000 + y^250000 ) ∙ ( x^250000 - y^250000 )
x^250000 - y^250000 = ( x^125000 )^2 - ( y^125000 )^2 =
( x^125000 + y^125000 ) ∙ ( x^125000 - y^125000 )
x^125000 - y^125000 = ( x^62500 )^2 - ( y^62500 )^2 =
( x^62500 + y^62500 ) ∙ ( x^62500 - y^62500 )
x^62500 - y^62500 = ( x^31250 )^2 - ( y^31250 )^2 =
( x^31250 + y^31250 ) ∙ ( x^31250 - y^31250 )
x^31250 - y^31250 = ( x^15625 )^2 - ( y^15625 )^2 =
( x^15625 + y^15625 ) ∙ ( x^15625 - y^15625 )
15625 can be further factorized because 15625 = 5^6