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?
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
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.
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
To determine the number of factors for a polynomial, we need to understand what factors are. In mathematics, a factor of a polynomial is an expression that can divide the polynomial evenly, leaving no remainder.
In the case of the polynomial x^1000000 - y^1000000, we can think of it as a difference of two terms, x^1000000 and y^1000000.
To find the number of factors for this polynomial, we can use the concept of factorization. Factorization involves breaking down a polynomial into its factors.
So, to find the factors of x^1000000 - y^1000000, we can focus on the algebraic identity (a^n - b^n) = (a - b)(a^(n-1) + a^(n-2)b + a^(n-3)b^2 + ... + ab^(n-2) + b^(n-1)).
By applying this identity, we can rewrite the polynomial as follows:
x^1000000 - y^1000000 = (x^1000)^1000 - (y^1000)^1000
= (x^1000 - y^1000)(x^999000 + x^998000y^1000 + x^997000y^2000 + ... + xy^999000 + y^1000000)
Here, we have factored the polynomial into two terms: (x^1000 - y^1000) and (x^999000 + x^998000y^1000 + x^997000y^2000 + ... + xy^999000 + y^1000000).
Now, the number of factors for the polynomial is equal to the number of factors of each term multiplied together. Therefore, to find the number of factors, we need to determine the number of factors for each of these terms separately.
The term (x^1000 - y^1000) represents a difference of squares, so it can be factored further using another identity: (a^2 - b^2) = (a - b)(a + b).
(x^1000 - y^1000) = (x^500 - y^500)(x^500 + y^500)
Now, the term (x^500 - y^500) is also a difference of squares and can be factored using the same identity as above.
Proceeding with this factorization process repeatedly, we eventually reach the factors of the polynomial.
To determine the actual number of factors, we count the total number of unique combinations of these factors. For example, if we have factors (a, b, c, d), there are 2^4 = 16 possible combinations. However, we need to exclude the combination with no factors and the combination with all factors, so the actual number of factors is 16 - 2 = 14.
Thus, by factoring the polynomial and counting the unique combinations of the factors, we can determine the number of factors for x^1000000 - y^1000000.