Answer :
To determine which expression is equivalent to [tex]\( (p - q)(x) \)[/tex], we need to start by computing [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex]. Given:
[tex]\[ p(x) = x^2 - 1 \][/tex]
[tex]\[ q(x) = 5(x - 1) \][/tex]
Next, we calculate [tex]\( (p - q)(x) \)[/tex]:
[tex]\[ (p - q)(x) = p(x) - q(x) \][/tex]
Substituting the given expressions for [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex]:
[tex]\[ (p - q)(x) = (x^2 - 1) - 5(x - 1) \][/tex]
Now, simplify the expression step-by-step:
1. Distribute the 5 in the term [tex]\( 5(x - 1) \)[/tex]:
[tex]\[ 5(x - 1) = 5x - 5 \][/tex]
2. Substitute this back into our main expression:
[tex]\[ (p - q)(x) = (x^2 - 1) - (5x - 5) \][/tex]
3. Simplify by distributing the negative sign through the second parenthesis:
[tex]\[ (p - q)(x) = x^2 - 1 - 5x + 5 \][/tex]
4. Combine like terms:
[tex]\[ (p - q)(x) = x^2 - 5x + 4 \][/tex]
Now, we need to match this simplified expression, [tex]\( x^2 - 5x + 4 \)[/tex], with one of the given choices.
Let's evaluate each given choice:
1. [tex]\( 5(x - 1) - x^2 - 1 \)[/tex]:
[tex]\[ 5(x - 1) - x^2 - 1 = 5x - 5 - x^2 - 1 = -x^2 + 5x - 6 \quad \text{(This is not equivalent to } x^2 - 5x + 4) \][/tex]
2. [tex]\( 5x - 1 - (x^2 - 1) \)[/tex]:
[tex]\[ 5x - 1 - (x^2 - 1) = 5x - 1 - x^2 + 1 = -x^2 + 5x \quad \text{(This is not equivalent to } x^2 - 5x + 4) \][/tex]
3. [tex]\( (x^2 - 1) - 5(x - 1) \)[/tex]:
[tex]\[ (x^2 - 1) - 5(x - 1) = (x^2 - 1) - 5x + 5 = x^2 - 5x + 4 \quad \text{(This matches our expression perfectly)} \][/tex]
4. [tex]\( (x^2 - 1) - 5x - 1 \)[/tex]:
[tex]\[ (x^2 - 1) - 5x - 1 = x^2 - 1 - 5x - 1 = x^2 - 5x - 2 \quad \text{(This is not equivalent to } x^2 - 5x + 4) \][/tex]
Thus, the correct expression equivalent to [tex]\( (p - q)(x) \)[/tex] is:
[tex]\[ \left(x^2-1\right)-5(x-1) \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{\left(x^2-1\right)-5(x-1)} \][/tex]
[tex]\[ p(x) = x^2 - 1 \][/tex]
[tex]\[ q(x) = 5(x - 1) \][/tex]
Next, we calculate [tex]\( (p - q)(x) \)[/tex]:
[tex]\[ (p - q)(x) = p(x) - q(x) \][/tex]
Substituting the given expressions for [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex]:
[tex]\[ (p - q)(x) = (x^2 - 1) - 5(x - 1) \][/tex]
Now, simplify the expression step-by-step:
1. Distribute the 5 in the term [tex]\( 5(x - 1) \)[/tex]:
[tex]\[ 5(x - 1) = 5x - 5 \][/tex]
2. Substitute this back into our main expression:
[tex]\[ (p - q)(x) = (x^2 - 1) - (5x - 5) \][/tex]
3. Simplify by distributing the negative sign through the second parenthesis:
[tex]\[ (p - q)(x) = x^2 - 1 - 5x + 5 \][/tex]
4. Combine like terms:
[tex]\[ (p - q)(x) = x^2 - 5x + 4 \][/tex]
Now, we need to match this simplified expression, [tex]\( x^2 - 5x + 4 \)[/tex], with one of the given choices.
Let's evaluate each given choice:
1. [tex]\( 5(x - 1) - x^2 - 1 \)[/tex]:
[tex]\[ 5(x - 1) - x^2 - 1 = 5x - 5 - x^2 - 1 = -x^2 + 5x - 6 \quad \text{(This is not equivalent to } x^2 - 5x + 4) \][/tex]
2. [tex]\( 5x - 1 - (x^2 - 1) \)[/tex]:
[tex]\[ 5x - 1 - (x^2 - 1) = 5x - 1 - x^2 + 1 = -x^2 + 5x \quad \text{(This is not equivalent to } x^2 - 5x + 4) \][/tex]
3. [tex]\( (x^2 - 1) - 5(x - 1) \)[/tex]:
[tex]\[ (x^2 - 1) - 5(x - 1) = (x^2 - 1) - 5x + 5 = x^2 - 5x + 4 \quad \text{(This matches our expression perfectly)} \][/tex]
4. [tex]\( (x^2 - 1) - 5x - 1 \)[/tex]:
[tex]\[ (x^2 - 1) - 5x - 1 = x^2 - 1 - 5x - 1 = x^2 - 5x - 2 \quad \text{(This is not equivalent to } x^2 - 5x + 4) \][/tex]
Thus, the correct expression equivalent to [tex]\( (p - q)(x) \)[/tex] is:
[tex]\[ \left(x^2-1\right)-5(x-1) \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{\left(x^2-1\right)-5(x-1)} \][/tex]
