Answer :
We start with the expression
[tex]$$
\frac{x^7 - x^\beta}{4x^9 - 4x^5}.
$$[/tex]
Step 1. Factor the Denominator
Factor out the common term in the denominator:
[tex]$$
4x^9 - 4x^5 = 4x^5(x^4 - 1).
$$[/tex]
Notice that the term [tex]$x^4-1$[/tex] is a difference of squares. It can be factored further:
[tex]$$
x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1).
$$[/tex]
Thus, the denominator may be written as
[tex]$$
4x^5(x-1)(x+1)(x^2+1).
$$[/tex]
Step 2. Factor the Numerator
The numerator is
[tex]$$
x^7 - x^\beta.
$$[/tex]
Since the exponent [tex]$\beta$[/tex] is not specified further, the numerator remains as
[tex]$$
x^7 - x^\beta.
$$[/tex]
Step 3. Write the Simplified Expression
After factoring, the expression becomes
[tex]$$
\frac{x^7 - x^\beta}{4x^5(x-1)(x+1)(x^2+1)}.
$$[/tex]
Alternatively, one may leave the denominator using the factorization [tex]$x^4 - 1$[/tex]:
[tex]$$
\frac{x^7 - x^\beta}{4x^5(x^4-1)}.
$$[/tex]
This is the simplified form of the original expression.
Final Answer
The original expression, its factorizations, and the simplified result are:
[tex]$$
\frac{x^7 - x^\beta}{4x^9-4x^5}, \quad x^7 - x^\beta, \quad 4x^5 (x-1)(x+1)(x^2+1), \quad \frac{x^7-x^\beta}{4x^5(x^4-1)}.
$$[/tex]
[tex]$$
\frac{x^7 - x^\beta}{4x^9 - 4x^5}.
$$[/tex]
Step 1. Factor the Denominator
Factor out the common term in the denominator:
[tex]$$
4x^9 - 4x^5 = 4x^5(x^4 - 1).
$$[/tex]
Notice that the term [tex]$x^4-1$[/tex] is a difference of squares. It can be factored further:
[tex]$$
x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1).
$$[/tex]
Thus, the denominator may be written as
[tex]$$
4x^5(x-1)(x+1)(x^2+1).
$$[/tex]
Step 2. Factor the Numerator
The numerator is
[tex]$$
x^7 - x^\beta.
$$[/tex]
Since the exponent [tex]$\beta$[/tex] is not specified further, the numerator remains as
[tex]$$
x^7 - x^\beta.
$$[/tex]
Step 3. Write the Simplified Expression
After factoring, the expression becomes
[tex]$$
\frac{x^7 - x^\beta}{4x^5(x-1)(x+1)(x^2+1)}.
$$[/tex]
Alternatively, one may leave the denominator using the factorization [tex]$x^4 - 1$[/tex]:
[tex]$$
\frac{x^7 - x^\beta}{4x^5(x^4-1)}.
$$[/tex]
This is the simplified form of the original expression.
Final Answer
The original expression, its factorizations, and the simplified result are:
[tex]$$
\frac{x^7 - x^\beta}{4x^9-4x^5}, \quad x^7 - x^\beta, \quad 4x^5 (x-1)(x+1)(x^2+1), \quad \frac{x^7-x^\beta}{4x^5(x^4-1)}.
$$[/tex]
