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Which expression is equivalent to [tex]\frac{2x^6 - 18x^4 + 2x^2}{2x^2}[/tex]?

A. [tex]x^3 - 9x^2[/tex]

B. [tex]x^4 - 9x^2[/tex]

C. [tex]x^3 - 9x^2 + 1[/tex]

D. [tex]x^4 - 9x^2 + 1[/tex]

Answer :

We start with the expression

[tex]$$
\frac{2 x^6 - 18 x^4 + 2 x^2}{2 x^2}.
$$[/tex]

Step 1. Factor the numerator

Notice that every term in the numerator has a common factor of [tex]$2x^2$[/tex]. Factor it out:

[tex]$$
2x^6 - 18x^4 + 2x^2 = 2x^2 \left( x^4 - 9x^2 + 1 \right).
$$[/tex]

Step 2. Cancel the common factor

Now, substitute the factored numerator back into the expression:

[tex]$$
\frac{2x^2 \left( x^4 - 9x^2 + 1 \right)}{2x^2}.
$$[/tex]

Since [tex]$2x^2$[/tex] is present in both the numerator and the denominator (and assuming [tex]$x \neq 0$[/tex] so that we can cancel), we cancel this common factor:

[tex]$$
\frac{2x^2 \left( x^4 - 9x^2 + 1 \right)}{2x^2} = x^4 - 9x^2 + 1.
$$[/tex]

Step 3. Match with the provided choices

The simplified expression is

[tex]$$
x^4 - 9x^2 + 1,
$$[/tex]

which corresponds to the option:

D. [tex]$x^4 - 9 x^2 + 1.$[/tex]

Thus, the equivalent expression is

[tex]$$
\boxed{x^4 - 9x^2 + 1}.
$$[/tex]

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