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Which expression is equivalent to the polynomial [tex]\(16x^2 + 4\)[/tex]?

A. [tex]\((4x + 2i)(4x - 2i)\)[/tex]
B. [tex]\((4x + 2)(4x - 2)\)[/tex]
C. [tex]\((4x + 2)^2\)[/tex]
D. [tex]\((4x - 2i)^2\)[/tex]

Sagot :

GinnyAnswer
To find an expression equivalent to the polynomial [tex]\( 16x^2 + 4 \)[/tex], let's go through the factoring process.

First, let's start from the given polynomial:

[tex]\[ 16x^2 + 4 \][/tex]

Notice that [tex]\( 16x^2 + 4 \)[/tex] can be factored by first identifying the common factor:

[tex]\[ 16x^2 + 4 = 4(4x^2 + 1) \][/tex]

Next, we need to factor [tex]\( 4x^2 + 1 \)[/tex]. This expression is a sum of squares, and can be written as:

[tex]\[ 4x^2 + 1 = (2x)^2 + (1)^2 \][/tex]

The sum of squares can be factored using complex numbers:

[tex]\[ a^2 + b^2 = (a + bi)(a - bi) \][/tex]

Here, [tex]\( a = 2x \)[/tex] and [tex]\( b = 1 \)[/tex]. Substituting these values into the factoring formula, we get:

[tex]\[ 4x^2 + 1 = (2x + i)(2x - i) \][/tex]

Now, substituting this back into our original expression:

[tex]\[ 16x^2 + 4 = 4((2x + i)(2x - i)) \][/tex]

Finally, note that we can further simplify [tex]\( 4((2x + i)(2x - i)) \)[/tex]:

[tex]\[ 4((2x + i)(2x - i)) = (4x + 2i)(4x - 2i) \][/tex]

Therefore, the expression equivalent to the polynomial [tex]\( 16x^2 + 4 \)[/tex] is:

[tex]\[ (4x + 2i)(4x - 2i) \][/tex]

So, the correct answer is:

[tex]\[ \boxed{A} \][/tex]
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