To solve the problem, we need to find the operation involving [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] that results in the target rational expression [tex]\(\frac{4}{x-1}\)[/tex]. Let's go step by step through each operation:
### Expressions Provided:
- [tex]\( P = \frac{5}{x-3} \)[/tex]
- [tex]\( Q = \frac{x+7}{x^2-4x+3} \)[/tex]
- Target result: [tex]\(\frac{4}{x-1}\)[/tex]
First, simplify the expression for [tex]\( Q \)[/tex]:
[tex]\[ Q = \frac{x+7}{x^2 - 4x + 3} \][/tex]
Factor the denominator [tex]\( x^2 - 4x + 3 \)[/tex]:
[tex]\[ x^2 - 4x + 3 = (x - 1)(x - 3) \][/tex]
So we have:
[tex]\[ Q = \frac{x+7}{(x-1)(x-3)} \][/tex]
### Option A: P - Q
[tex]\[ P - Q = \frac{5}{x-3} - \frac{x+7}{(x-1)(x-3)} \][/tex]
To subtract these rational expressions, find a common denominator:
[tex]\[ P - Q = \frac{5(x-1) - (x+7)}{(x-1)(x-3)} \][/tex]
Expand the numerator:
[tex]\[ P - Q = \frac{5x - 5 - x - 7}{(x-1)(x-3)} \][/tex]
[tex]\[ P - Q = \frac{4x - 12}{(x-1)(x-3)} \][/tex]
Factor out the common term in the numerator:
[tex]\[ P - Q = \frac{4(x - 3)}{(x-1)(x-3)} \][/tex]
Cancel out [tex]\((x-3)\)[/tex]:
[tex]\[ P - Q = \frac{4}{x-1} \][/tex]
This matches the target rational expression [tex]\(\frac{4}{x-1}\)[/tex]. Therefore, the correct operation is subtracting [tex]\( Q \)[/tex] from [tex]\( P \)[/tex].
So, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
### Verification of Other Options (briefly):
#### Option B: P + Q
[tex]\[ P + Q = \frac{5}{x-3} + \frac{x+7}{(x-1)(x-3)} \][/tex]
Combine with a common denominator:
[tex]\[ P + Q = \frac{5(x-1) + (x+7)}{(x-1)(x-3)} \][/tex]
Expand and simplify the numerator:
[tex]\[ P + Q = \frac{5x - 5 + x + 7}{(x-1)(x-3)} \][/tex]
[tex]\[ P + Q = \frac{6x + 2}{(x-1)(x-3)} \][/tex]
This does not simplify to [tex]\(\frac{4}{x-1}\)[/tex], so option B is incorrect.
#### Option C: P \cdot Q
[tex]\[ P \times Q = \frac{5}{x-3} \times \frac{x+7}{(x-1)(x-3)} \][/tex]
[tex]\[ P \times Q = \frac{5(x+7)}{(x-3)^2 (x-1)} \][/tex]
This does not simplify to [tex]\(\frac{4}{x-1}\)[/tex], so option C is incorrect.
#### Option D: P \div Q
[tex]\[ P \div Q = \frac{5}{x-3} \div \frac{x+7}{(x-1)(x-3)} \][/tex]
[tex]\[ P \div Q = \frac{5}{x-3} \times \frac{(x-1)(x-3)}{x+7} \][/tex]
[tex]\[ P \div Q = \frac{5(x-1)}{x+7} \][/tex]
This does not simplify to [tex]\(\frac{4}{x-1}\)[/tex], so option D is incorrect.
Hence, option A is indeed the correct operation.
