Question

If p(x) = x2 – 1 and q (x) = 5 (x minus 1), which expression is equivalent to (p – q)(x)?
5(x – 1) – x2 – 1
(5x – 1) – (x2 – 1)
(x2 – 1) – 5(x – 1)
(x2 – 1) – 5x – 1

Answer 1

We have the functions \( p(x) = x^2 - 1 \) and \( q(x) = 5(x - 1) \).

To find the expression for \( (p - q)(x) \), we need to compute \( p(x) - q(x) \).

First, we can compute \( q(x) \):

\[
q(x) = 5(x - 1) = 5x - 5
\]

Now we can substitute \( p(x) \) and \( q(x) \) into \( (p - q)(x) \):

\[
(p - q)(x) = p(x) - q(x) = (x^2 - 1) - (5x - 5)
\]

Now, we will simplify this expression:

\[
(p - q)(x) = x^2 - 1 - 5x + 5
\]
\[
= x^2 - 5x + 4
\]

Now we can compare this with the options given:

1. \( 5(x - 1) - x^2 - 1 \)
2. \( (5x - 1) - (x^2 - 1) \)
3. \( (x^2 - 1) - 5(x - 1) \)
4. \( (x^2 - 1) - 5x - 1 \)

Now let's check \( (p - q)(x) = x^2 - 5x + 4 \) against the options:

1. **For Option 1:**
\[
5(x - 1) - (x^2 - 1) = 5x - 5 - x^2 + 1 = -x^2 + 5x - 4
\]

2. **For Option 2:**
\[
(5x - 1) - (x^2 - 1) = 5x - 1 - x^2 + 1 = -x^2 + 5x
\]

3. **For Option 3:**
\[
(x^2 - 1) - 5(x - 1) = x^2 - 1 - 5x + 5 = x^2 - 5x + 4
\]

4. **For Option 4:**
\[
(x^2 - 1) - 5x - 1 = x^2 - 1 - 5x - 1 = x^2 - 5x - 2
\]

The expression that matches \( (p - q)(x) = x^2 - 5x + 4 \) is the third option:

**Answer: (x^2 - 1) - 5(x - 1)**