To find the values of [tex]\( P(0) \)[/tex], [tex]\( P(2) \)[/tex], and [tex]\( P(\pi) \)[/tex] for the polynomial [tex]\( P(x) = (x + 1)(x - 1) \)[/tex], follow the steps below:
Step 1: Understand the polynomial P(x)
The given polynomial is:
[tex]\[ P(x) = (x + 1)(x - 1). \][/tex]
This can also be expressed as:
[tex]\[ P(x) = x^2 - 1, \][/tex]
using the difference of squares formula.
Step 2: Calculate [tex]\( P(0) \)[/tex]
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( P(x) \)[/tex]:
[tex]\[ P(0) = (0 + 1)(0 - 1) = 1 \cdot (-1) = -1. \][/tex]
So, [tex]\( P(0) = -1 \)[/tex].
Step 3: Calculate [tex]\( P(2) \)[/tex]
Substitute [tex]\( x = 2 \)[/tex] into [tex]\( P(x) \)[/tex]:
[tex]\[ P(2) = (2 + 1)(2 - 1) = 3 \cdot 1 = 3. \][/tex]
So, [tex]\( P(2) = 3 \)[/tex].
Step 4: Calculate [tex]\( P(\pi) \)[/tex]
Substitute [tex]\( x = \pi \)[/tex] into [tex]\( P(x) \)[/tex]:
[tex]\[ P(\pi) = (\pi + 1)(\pi - 1). \][/tex]
First, approximate [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[ \pi + 1 \approx 4.14159 \][/tex]
[tex]\[ \pi - 1 \approx 2.14159 \][/tex]
Then, multiply the approximate values:
[tex]\[ P(\pi) \approx 4.14159 \times 2.14159 \approx 8.8696. \][/tex]
So, [tex]\( P(\pi) \approx 8.87 \)[/tex] (rounded to 2 decimal places for practical purposes, but keeping in mind the accurate result is [tex]\( 8.869604401089358 \)[/tex]).
Summary of Results:
- [tex]\( P(0) = -1 \)[/tex]
- [tex]\( P(2) = 3 \)[/tex]
- [tex]\( P(\pi) \approx 8.87 \)[/tex] (or more precisely, [tex]\( 8.869604401089358 \)[/tex])
These are the values of the polynomial [tex]\( P(x) = (x + 1)(x - 1) \)[/tex] evaluated at [tex]\( x = 0 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = \pi \)[/tex].
