Sure, let's find the expression equivalent to [tex]\((p - q)(x)\)[/tex] by following these steps:
1. Substitute the expressions for [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex]:
- Given [tex]\( p(x) = x^2 - 1 \)[/tex]
- Given [tex]\( q(x) = 5(x - 1) \)[/tex]
2. Set up the expression for [tex]\((p - q)(x)\)[/tex]:
[tex]\[
(p - q)(x) = p(x) - q(x)
\][/tex]
Substitute [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex]:
[tex]\[
(p - q)(x) = (x^2 - 1) - 5(x - 1)
\][/tex]
3. Simplify the expression:
- Start by distributing [tex]\( 5 \)[/tex] in the [tex]\( q(x) \)[/tex] term:
[tex]\[
5(x - 1) = 5x - 5
\][/tex]
- Substitute back into the expression:
[tex]\[
(p - q)(x) = (x^2 - 1) - (5x - 5)
\][/tex]
- Distribute the negative sign:
[tex]\[
(p - q)(x) = x^2 - 1 - 5x + 5
\][/tex]
- Combine like terms:
[tex]\[
(p - q)(x) = x^2 - 5x + 4
\][/tex]
4. Compare with the given options:
- [tex]\(5(x-1)-x^2-1\)[/tex]
- [tex]\((5x-1)-\left(x^2-1\right)\)[/tex]
- [tex]\(\left(x^2-1\right)-5(x-1)\)[/tex]
- [tex]\(\left(x^2-1\right)-5x-1\)[/tex]
Option three, [tex]\(\left(x^2 - 1\right) - 5(x - 1)\)[/tex], matches our derived expression.
Hence, the equivalent expression to [tex]\((p - q)(x)\)[/tex] is:
[tex]\[
\boxed{\left(x^2-1\right)-5(x-1)}
\][/tex]
