Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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]

To find the polynomial expression that factors to [tex]$x^4(4x-7)(4x+7)$[/tex], let's analyze what the factorization means and expand it.

The expression [tex]$x^4(4x-7)(4x+7)$[/tex] involves two binomials multiplied together:
1. [tex]$(4x-7)$[/tex]
2. [tex]$(4x+7)$[/tex]

Notice that [tex]$(4x-7)(4x+7)$[/tex] follows the pattern of a "difference of squares," which is generally expressed as:

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

In this case, [tex]$a = 4x$[/tex] and [tex]$b = 7$[/tex], so:

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

Now, multiply this by [tex]$x^4$[/tex]:

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

Therefore, we are looking for the polynomial expression that factors to [tex]$16x^6 - 49x^4$[/tex].

Examining 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]

The correct expression, which matches [tex]$16x^6 - 49x^4$[/tex], is option 3:

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

So, the correct factorization corresponds to option 3: [tex]$16x^6 - 49x^4$[/tex].