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Sagot :
Certainly! Let's simplify the given mathematical expression step by step:
Given expression:
[tex]\[ \frac{1}{q+1} - \frac{q}{q^2 - 1} - \frac{q^2}{1 - q^4} \][/tex]
### Step 1: Identify common factors and simplify each term
First, recognize the factorizations in the denominators:
- [tex]\( q^2 - 1 \)[/tex] can be factored as [tex]\( (q - 1)(q + 1) \)[/tex]
- [tex]\( 1 - q^4 \)[/tex] can be factored as [tex]\( (1 - q^2)(1 + q^2) \)[/tex], which in turn can be factored further since [tex]\( 1 - q^2 \)[/tex] is [tex]\( (1 - q)(1 + q) \)[/tex]. So, [tex]\( 1 - q^4 = (1 - q)(1 + q)(1 + q^2) \)[/tex].
Thus, the expression becomes:
[tex]\[ \frac{1}{q+1} - \frac{q}{(q-1)(q+1)} - \frac{q^2}{(1 - q)(1 + q)(1 + q^2)} \][/tex]
### Step 2: Simplify each individual fraction
#### First Term:
[tex]\[ \frac{1}{q+1} \][/tex]
#### Second Term:
[tex]\[ \frac{q}{(q-1)(q+1)} \][/tex]
We notice this can be simplified:
[tex]\[ \frac{q}{(q-1)(q+1)} = \frac{q}{(q^2 - 1)} \][/tex]
#### Third Term:
[tex]\[ \frac{q^2}{(1 - q)(1 + q)(1 + q^2)} \][/tex]
We notice [tex]\( (1 - q) = -(q - 1) \)[/tex] and thus rewrite it:
[tex]\[ \frac{q^2}{(q - 1)(q + 1)(q^2 + 1)} \][/tex]
### Step 3: Combine the simplified fractions
Now, we need to combine these back together:
[tex]\[ \frac{1}{q+1} - \frac{q}{q^2 - 1} - \frac{q^2}{(q - 1)(q + 1)(q^2 + 1)} \][/tex]
Given that [tex]\( q^2 - 1 = (q - 1)(q + 1) \)[/tex], we can write the expression with a common denominator:
[tex]\[ = \frac{1}{q + 1} - \frac{q}{(q - 1)(q + 1)} - \frac{q^2}{(q - 1)(q + 1)(q^2 + 1)} \][/tex]
### Step 4: Find a common denominator
The common denominator for all fractions is [tex]\( (q + 1)(q - 1)(q^2 + 1) \)[/tex]:
[tex]\[ \frac{(q - 1)(q^2 + 1)}{(q + 1)(q - 1)(q^2 + 1)} - \frac{q(q^2 + 1)}{(q + 1)(q - 1)(q^2 + 1)} - \frac{q^2}{(q + 1)(q - 1)(q^2 + 1)} \][/tex]
Now, simplify the numerators step by step.
Numerator after common denominator:
[tex]\[ (q - 1)(q^2 + 1) - q(q^2 + 1) - q^2 \][/tex]
### Step 5: Expand and simplify the numerator
Expand [tex]\( (q - 1)(q^2 + 1) \)[/tex]:
[tex]\[ q^3 + q - q^2 - 1 \][/tex]
So, we have:
[tex]\[ (q^3 + q - q^2 - 1) - (q^3 + q) - q^2 \][/tex]
[tex]\[ = q^3 + q - q^2 - 1 - q^3 - q - q^2 \][/tex]
[tex]\[ = -q^2 - q^2 - 1 + q - q - q \][/tex]
[tex]\[ = -2q^2 - 1 \][/tex]
### Step 6: Combine and simplify
This results in:
[tex]\[ - \frac{2q^2 + 1}{(q + 1)(q - 1)(q^2 + 1)} \][/tex]
Since [tex]\( q^4 - 1 = (q^2 - 1)(q^2 + 1) = (q - 1)(q + 1)(q^2 + 1) \)[/tex]:
[tex]\[ - \frac{1}{q^4 - 1} \][/tex]
Thus, the simplified expression is:
[tex]\[ - \frac{1}{q^4 - 1} \][/tex]
So, our final simplified expression is:
[tex]\[ -\frac{1}{q^4 - 1} \][/tex]
This is the simplified form of the given expression.
Given expression:
[tex]\[ \frac{1}{q+1} - \frac{q}{q^2 - 1} - \frac{q^2}{1 - q^4} \][/tex]
### Step 1: Identify common factors and simplify each term
First, recognize the factorizations in the denominators:
- [tex]\( q^2 - 1 \)[/tex] can be factored as [tex]\( (q - 1)(q + 1) \)[/tex]
- [tex]\( 1 - q^4 \)[/tex] can be factored as [tex]\( (1 - q^2)(1 + q^2) \)[/tex], which in turn can be factored further since [tex]\( 1 - q^2 \)[/tex] is [tex]\( (1 - q)(1 + q) \)[/tex]. So, [tex]\( 1 - q^4 = (1 - q)(1 + q)(1 + q^2) \)[/tex].
Thus, the expression becomes:
[tex]\[ \frac{1}{q+1} - \frac{q}{(q-1)(q+1)} - \frac{q^2}{(1 - q)(1 + q)(1 + q^2)} \][/tex]
### Step 2: Simplify each individual fraction
#### First Term:
[tex]\[ \frac{1}{q+1} \][/tex]
#### Second Term:
[tex]\[ \frac{q}{(q-1)(q+1)} \][/tex]
We notice this can be simplified:
[tex]\[ \frac{q}{(q-1)(q+1)} = \frac{q}{(q^2 - 1)} \][/tex]
#### Third Term:
[tex]\[ \frac{q^2}{(1 - q)(1 + q)(1 + q^2)} \][/tex]
We notice [tex]\( (1 - q) = -(q - 1) \)[/tex] and thus rewrite it:
[tex]\[ \frac{q^2}{(q - 1)(q + 1)(q^2 + 1)} \][/tex]
### Step 3: Combine the simplified fractions
Now, we need to combine these back together:
[tex]\[ \frac{1}{q+1} - \frac{q}{q^2 - 1} - \frac{q^2}{(q - 1)(q + 1)(q^2 + 1)} \][/tex]
Given that [tex]\( q^2 - 1 = (q - 1)(q + 1) \)[/tex], we can write the expression with a common denominator:
[tex]\[ = \frac{1}{q + 1} - \frac{q}{(q - 1)(q + 1)} - \frac{q^2}{(q - 1)(q + 1)(q^2 + 1)} \][/tex]
### Step 4: Find a common denominator
The common denominator for all fractions is [tex]\( (q + 1)(q - 1)(q^2 + 1) \)[/tex]:
[tex]\[ \frac{(q - 1)(q^2 + 1)}{(q + 1)(q - 1)(q^2 + 1)} - \frac{q(q^2 + 1)}{(q + 1)(q - 1)(q^2 + 1)} - \frac{q^2}{(q + 1)(q - 1)(q^2 + 1)} \][/tex]
Now, simplify the numerators step by step.
Numerator after common denominator:
[tex]\[ (q - 1)(q^2 + 1) - q(q^2 + 1) - q^2 \][/tex]
### Step 5: Expand and simplify the numerator
Expand [tex]\( (q - 1)(q^2 + 1) \)[/tex]:
[tex]\[ q^3 + q - q^2 - 1 \][/tex]
So, we have:
[tex]\[ (q^3 + q - q^2 - 1) - (q^3 + q) - q^2 \][/tex]
[tex]\[ = q^3 + q - q^2 - 1 - q^3 - q - q^2 \][/tex]
[tex]\[ = -q^2 - q^2 - 1 + q - q - q \][/tex]
[tex]\[ = -2q^2 - 1 \][/tex]
### Step 6: Combine and simplify
This results in:
[tex]\[ - \frac{2q^2 + 1}{(q + 1)(q - 1)(q^2 + 1)} \][/tex]
Since [tex]\( q^4 - 1 = (q^2 - 1)(q^2 + 1) = (q - 1)(q + 1)(q^2 + 1) \)[/tex]:
[tex]\[ - \frac{1}{q^4 - 1} \][/tex]
Thus, the simplified expression is:
[tex]\[ - \frac{1}{q^4 - 1} \][/tex]
So, our final simplified expression is:
[tex]\[ -\frac{1}{q^4 - 1} \][/tex]
This is the simplified form of the given expression.
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