Understanding the Expression
To determine if the product of \( b + \sqrt{3} \) and \( 2 - \sqrt{3} \) is an irrational number, we start by calculating the product:
\[
(b + \sqrt{3})(2 - \sqrt{3})
\]
Using the distributive property (FOIL method):
Step 1: Use FOIL
- First: \( b \cdot 2 = 2b \)
- Outside: \( b \cdot (-\sqrt{3}) = -b\sqrt{3} \)
- Inside: \( \sqrt{3} \cdot 2 = 2\sqrt{3} \)
- Last: \( \sqrt{3} \cdot (-\sqrt{3}) = -3 \)
Step 2: Combine Terms
Combining these results gives:
\[
2b + (-b\sqrt{3} + 2\sqrt{3}) - 3
\]
This simplifies to:
\[
2b - 3 + (2 - b)\sqrt{3}
\]
Analyzing the Result
The expression \( 2b - 3 + (2 - b)\sqrt{3} \) consists of a rational part \( 2b - 3 \) and an irrational part \( (2 - b)\sqrt{3} \).
Determining Irrationality
- The product will be rational only if the coefficient of \( \sqrt{3} \) is zero, which occurs when \( 2 - b = 0 \) or \( b = 2 \).
- If \( b \) is not equal to 2, the term \( (2 - b)\sqrt{3} \) remains irrational.
Conclusion
Thus, the statement that the product is an irrational number is True unless \( b = 2 \). Therefore, the product is irrational for most values of \( b \).