Which expression is equivalent to this polynomial?
$16 x^2+4$
The goal is to determine the equivalent expression for the polynomial \(16x^2 + 4\). We examine this expression and recognize it as a sum of squares, which can be factored using the properties of complex numbers. Specifically, since the square of the imaginary unit \(i\) is \(i^2 = -1\), we can rewrite the addition as a subtraction: \(16x^2 + 4 = 16x^2 - (-4) = 16x^2 - (2i)^2\).
The expression \(16x^2 - (2i)^2\) is now in the form of a difference of squares, \(a^2 - b^2\), where \(a = 4x\) and \(b = 2i\). The difference of squares identity is given by:
\[a^2 - b^2 = (a - b)(a + b)\]
Applying this identity with \(a = 4x\) and \(b = 2i\), we get:
\[(4x)^2 - (2i)^2 = (4x - 2i)(4x + 2i)\]
Comparing this result with the provided options, we see that it matches the first option.
To be absolutely sure, we can expand each given option to verify which one results in the original polynomial \(16x^2 + 4\):
Thus, the correct expression is \((4x + 2i)(4x - 2i)\).
Understanding how to factor sums of squares is very useful in electrical engineering and signal processing. In these fields, complex numbers are used to represent oscillating signals, and factoring expressions like these helps engineers find the roots of characteristic equations, which describe how systems behave over time. Being able to manipulate these algebraic expressions allows for the simplification of complex circuits and systems analysis.