if both x-2 and `x-(1)/(2)` are factors of `px^(2)+5x+r.` then show that p=r.

Explanation

To solve the problem, we need to show that if both \( x - 2 \) and \( x - \frac{1}{2} \) are factors of the polynomial \( px^2 + 5x + r \), then \( p = r \).

### Step-by-Step Solution:

1. **Identify the polynomial**: We have the polynomial \( f(x) = px^2 + 5x + r \).

2. **Use the factor theorem**: Since \( x - 2 \) is a factor, we can substitute \( x = 2 \) into the polynomial and set it equal to zero:
\[
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