Let’s analyze the given polynomial [tex]\(16x^2 + 4\)[/tex] and find the correct factored form.
1. Identify the structure of the polynomial:
The polynomial [tex]\(16x^2 + 4\)[/tex] can be rewritten by recognizing it as a sum of squares. Notice that:
[tex]\[
16x^2 = (4x)^2 \quad \text{and} \quad 4 = 2^2
\][/tex]
So, we can rewrite the polynomial as:
[tex]\[
(4x)^2 + 2^2
\][/tex]
2. Consider the provided options:
- Option A: [tex]\((4x + 2i)(4x - 2i)\)[/tex]:
This expression involves imaginary numbers because of the presence of [tex]\(i\)[/tex], which is the imaginary unit. This suggests that the polynomial would be factored over the complex numbers and it corresponds to a format for a sum of squares but with imaginary numbers involved. However, we aim to identify a solution over real numbers if possible.
- Option B: [tex]\((4x + 2)(4x - 2)\)[/tex]:
This is essentially a difference of squares form:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
However, our given polynomial is a sum, not a difference. Let’s see:
[tex]\[
(4x + 2)(4x - 2) = 16x^2 - 4
\][/tex]
This expression simplifies to [tex]\(16x^2 - 4\)[/tex], which is not the same as our original polynomial. So this does not correspond.
- Option C: [tex]\((4x + 2)^2\)[/tex]:
This represents a perfect square:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
Let’s verify:
[tex]\[
(4x + 2)^2 = (4x)^2 + 2 \cdot 4x \cdot 2 + 2^2 = 16x^2 + 16x + 4
\][/tex]
This expression does not match our original polynomial either, as it includes a [tex]\(16x\)[/tex] term that is not present in [tex]\(16x^2 + 4\)[/tex].
- Option D: [tex]\((4x - 2i)^2\)[/tex]:
Similar to option A, this involves imaginary numbers ([tex]\(i\)[/tex]). This would represent a sum of squares over the complex field:
[tex]\[
(4x - 2i)^2 = (4x)^2 - 2\cdot 4x \cdot 2i + (2i)^2 = 16x^2 - 16xi^2 + (2i)^2
\][/tex]
This approach involves imaginary components and would translate to complex roots which are not factored over the real numbers correctly.
Considering all the options and the structure of the polynomial [tex]\(16x^2 + 4\)[/tex], the closest correct factored form without involving imaginary numbers is:
- Option B: [tex]\((4x + 2)(4x - 2)\)[/tex]
So the correct answer is:
[tex]\[
B. \quad (4x + 2)(4x - 2)
\][/tex]
