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Factor completely [tex]\(16 x^4-81\)[/tex].

A. [tex]\((2x - 3)(2x - 3)(4x^2 + 9)\)[/tex]

B. [tex]\((2x - 3)(2x + 3)(4x^2 + 9)\)[/tex]

C. [tex]\((2x - 3)(2x + 3)(4x^2 - 9)\)[/tex]

D. [tex]\((2x + 3)(2x + 3)(4x^2 + 9)\)[/tex]


Sagot :

To factor the expression [tex]\(16 x^4 - 81\)[/tex] completely, we'll go through the following steps:

1. Recognize the difference of squares pattern: The given expression [tex]\(16 x^4 - 81\)[/tex] can be seen as a difference of two squares. Recall that the difference of squares formula is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].

Here, we can rewrite [tex]\(16 x^4 - 81\)[/tex] as:
[tex]\[ (4x^2)^2 - 9^2 \][/tex]

2. Apply the difference of squares formula: The difference of squares formula can be applied to [tex]\( (4x^2)^2 - 9^2 \)[/tex] as follows:
[tex]\[ 16 x^4 - 81 = (4x^2 - 9)(4x^2 + 9) \][/tex]

3. Factor [tex]\(4x^2 - 9\)[/tex] further: Notice that [tex]\(4x^2 - 9\)[/tex] itself is another difference of squares because [tex]\(4x^2\)[/tex] and [tex]\(9\)[/tex] can both be written as squares:
[tex]\[ 4x^2 - 9 = (2x)^2 - 3^2 \][/tex]

Applying the difference of squares formula again, we get:
[tex]\[ 4x^2 - 9 = (2x - 3)(2x + 3) \][/tex]

4. Combine all factors: Substituting back into the original expression, we combine all the factors together:
[tex]\[ 16 x^4 - 81 = (2x - 3)(2x + 3)(4x^2 + 9) \][/tex]

So, the completely factored form of [tex]\(16 x^4 - 81\)[/tex] is:
[tex]\[ (2 x-3)(2 x+3)\left(4 x^2+9\right) \][/tex]

Thus, the correct choice among the options provided is:
[tex]\[ (2 x-3)(2 x+3)\left(4 x^2+9\right) \][/tex]