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Factor the polynomial completely:

[tex]x^4 - 2x^2 - 35[/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. [tex]x^4 - 2x^2 - 35 = \square[/tex] (Factor completely.)
B. [tex]x^4 - 2x^2 - 35[/tex] is prime.

Answer :

To factor the polynomial [tex]\(x^4 - 2x^2 - 35\)[/tex] completely, we begin by recognizing it as a quadratic in form of [tex]\(y^2 - 2y - 35\)[/tex], where [tex]\(y = x^2\)[/tex].

1. Substitution Step: Set [tex]\(y = x^2\)[/tex], then express the polynomial as:
[tex]\[
y^2 - 2y - 35
\][/tex]

2. Factor the Quadratic: We need two numbers that multiply to [tex]\(-35\)[/tex] and add to [tex]\(-2\)[/tex]. These numbers are [tex]\(5\)[/tex] and [tex]\(-7\)[/tex].

3. Write as a Product:
[tex]\[
y^2 - 2y - 35 = (y - 7)(y + 5)
\][/tex]

4. Replace [tex]\(y\)[/tex] back with [tex]\(x^2\)[/tex]:
[tex]\[
(y - 7)(y + 5) \quad \Rightarrow \quad (x^2 - 7)(x^2 + 5)
\][/tex]

So, the polynomial [tex]\(x^4 - 2x^2 - 35\)[/tex] factors completely as [tex]\((x^2 - 7)(x^2 + 5)\)[/tex].

Therefore, the correct choice is:
A. [tex]\(x^4-2x^2-35 = (x^2 - 7)(x^2 + 5)\)[/tex] (Factor completely.)

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