To find the expression that is equivalent to [tex]\( (p - q)(x) \)[/tex], we need to subtract [tex]\( q(x) \)[/tex] from [tex]\( p(x) \)[/tex].
Given:
[tex]\[ p(x) = x^2 - 1 \][/tex]
[tex]\[ q(x) = 5(x - 1) \][/tex]
We need to compute [tex]\( (p - q)(x) \)[/tex]:
[tex]\[ (p - q)(x) = p(x) - q(x) \][/tex]
Substitute the expressions for [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex]:
[tex]\[ (p - q)(x) = (x^2 - 1) - 5(x - 1) \][/tex]
Now, let's simplify this expression step-by-step:
1. Distribute the -5 across the terms inside the parentheses:
[tex]\[ 5(x - 1) = 5x - 5 \][/tex]
Thus:
[tex]\[ -5(x - 1) = -5x + 5 \][/tex]
2. Now substitute this back into the original expression:
[tex]\[ (p - q)(x) = (x^2 - 1) - 5x + 5 \][/tex]
3. Combine like terms:
[tex]\[ (p - q)(x) = x^2 - 1 - 5x + 5 \][/tex]
[tex]\[ (p - q)(x) = x^2 - 5x + 4 \][/tex]
Comparing this to the given options:
- Option 1: [tex]\( 5(x - 1) - x^2 - 1 \)[/tex] is not correct.
- Option 2: [tex]\( (5x - 1) - (x^2 - 1) \)[/tex] is not correct.
- Option 3: [tex]\( (x^2 - 1) - 5(x - 1) \)[/tex] simplifies correctly to [tex]\( x^2 - 5x + 4 \)[/tex].
- Option 4: [tex]\( (x^2 - 1) - 5x - 1 \)[/tex] is not correct.
The expression is correctly simplified in option 3.
Thus, the equivalent expression is:
[tex]\[ \boxed{(x^2 - 1) - 5(x - 1)} \][/tex]
