- Rewrite the polynomial \(x^2 + 12\) as a difference of squares: \(x^2 - (-12) = x^2 - (2\sqrt{3}i)^2\).
- Apply the difference of squares identity \(a^2 - b^2 = (a - b)(a + b)\) to factor the expression into \((x + 2\sqrt{3}i)(x - 2\sqrt{3}i)\).
- Confirm that the result matches the first option.
- The equivalent expression is \(\boxed{(x + 2\sqrt{3}i)(x - 2\sqrt{3}i)}\).
Explanation
- Problem Analysis
We are presented with the polynomial expression \(x^2 + 12\). Our goal is to identify which of the provided options is equivalent to this polynomial. To do this, we can utilize the algebraic identity for the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\). Although the given expression is a sum of squares, we can manipulate it into a difference of squares by using the imaginary unit \(i\), where \(i^2 = -1\). This will allow us to rewrite the expression as \(x^2 - (-12)\).
- Factoring the Polynomial
To rewrite \(x^2 + 12\) as a difference of squares, we need to express \(12\) as the square of some term multiplied by \(-1\). We know that \(-1 = i^2\). Therefore, \(-12 = 12 \cdot i^2\). Taking the square root of 12, we find \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\). Thus, we can write:
\[12 = (2\sqrt{3})^2\]
\[-12 = -(2\sqrt{3})^2 = (2\sqrt{3}i)^2\]
Now, we rewrite our original polynomial:
\[x^2 + 12 = x^2 - (-12) = x^2 - (2\sqrt{3}i)^2\]
Applying the difference of squares identity \(a^2 - b^2 = (a - b)(a + b)\) with \(a = x\) and \(b = 2\sqrt{3}i\), we get:
\[x^2 - (2\sqrt{3}i)^2 = (x - 2\sqrt{3}i)(x + 2\sqrt{3}i)\]
This is equivalent to the first option provided.
- Verification and Conclusion
Let's verify the other options to ensure our conclusion is correct:
- \((x + 2\sqrt{3}i)(x - 2\sqrt{3}i) = x^2 - (2\sqrt{3}i)^2 = x^2 - (12 \cdot -1) = x^2 + 12\). This is correct.
- \((x + 6i)(x - 6i) = x^2 - (6i)^2 = x^2 - 36(-1) = x^2 + 36\). This is incorrect.
- \((x + 2\sqrt{3})^2 = x^2 + 4\sqrt{3}x + 12\). This is incorrect.
- \((x + 2\sqrt{3})(x - 2\sqrt{3}) = x^2 - (2\sqrt{3})^2 = x^2 - 12\). This is incorrect.
Therefore, the equivalent expression is \((x + 2\sqrt{3}i)(x - 2\sqrt{3}i)\).
Examples
Understanding how to factor expressions involving the sum of squares is essential in fields like electrical engineering and signal processing. For instance, when analyzing AC circuits, engineers often work with complex numbers to represent impedance. Expressing a sum of squares as a product of complex factors, like factoring \(x^2 + \omega^2\) as \((x + \omega i)(x - \omega i)\), helps simplify complex equations and solve for resonance frequencies, which are critical for designing efficient communication systems and power grids.