Answered

Find expert answers and community insights on IDNLearn.com. Get comprehensive answers to all your questions from our network of experienced experts.

Select the correct answer.

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 :

GinnyAnsweridn
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]
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Your questions find clarity at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.