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------------------------------------------------ Which of the following polynomial expressions factors to [tex]$x^4(4x-7)(4x+7)$[/tex]?

A. [tex]4x^6 - 7x^4[/tex]

B. [tex]16x^4 - 49x^2[/tex]

C. [tex]16x^6 - 49x^4[/tex]

D. [tex]16x^6 + 49x^4[/tex]

Answer :

To solve the problem of finding which polynomial expression factors to [tex]\( x^4(4x-7)(4x+7) \)[/tex], we need to identify the polynomial that, when factored, matches this expression.

First, observe that [tex]\( (4x-7)(4x+7) \)[/tex] is a difference of squares:

[tex]\[
(4x-7)(4x+7) = (4x)^2 - 7^2 = 16x^2 - 49
\][/tex]

So, the expression [tex]\( x^4(4x-7)(4x+7) \)[/tex] when expanded, becomes:

[tex]\[
x^4 \cdot (16x^2 - 49)
\][/tex]

Distribute [tex]\( x^4 \)[/tex]:

[tex]\[
x^4 \cdot 16x^2 - x^4 \cdot 49 = 16x^6 - 49x^4
\][/tex]

Thus, we are looking for the polynomial expression that expands to [tex]\( 16x^6 - 49x^4 \)[/tex].

Now, let's check the given options:

1. [tex]\( 4x^6 - 7x^4 \)[/tex]
2. [tex]\( 16x^4 - 49x^2 \)[/tex]
3. [tex]\( 16x^6 - 49x^4 \)[/tex]
4. [tex]\( 16x^6 + 49x^4 \)[/tex]

From the analysis above, we can see that the correct expression is [tex]\( 16x^6 - 49x^4 \)[/tex], which matches option 3.

Therefore, the correct answer is:

[tex]\( 16x^6 - 49x^4 \)[/tex]