Respuesta :

xKelvin

Answer:

The answer should be:

[tex]=\frac{8x^3-18x}{3x+2}[/tex]

The closest answer is A, so I'll go with that.

Step-by-step explanation:

So we have the rational expression:

[tex]\frac{4x^3-9x}{2x-7}\div\frac{3x^2+2x^2}{4x^2-14x}[/tex]

First, remove the division sign. To do so, turn the division into multiplication and flip the second term:

[tex]\frac{4x^3-9x}{2x-7}\cdot \frac{4x^2-14x}{3x^2+2x}[/tex]

Now, simplify. From the first term, in the numerator, factor out a x. On the second term, factor out a 2x in the numerator and a x in the denominator:

[tex]\frac{x(4x^2-9)}{2x-7}\cdot\frac{2x(2x-7)}{x(3x+2)}[/tex]

Multiply straight across:

[tex]\frac{x(4x^2-9)(2x)(2x-7)}{(2x-7)(x)(3x+2)}[/tex]

Cancel out the (2x-7):

[tex]\frac{x(4x^2-9)(2x)}{x(3x+2)}[/tex]

Cancel out the x:

[tex]\frac{(2x)(4x^2-9)}{(3x+2)}[/tex]

At this point, we can factor the (4x^2-9) term, but we won't be able to cancel it out. Thus, this is the simplest it can get.

To get the answer, expand the numerator:

[tex]=\frac{8x^3-18x}{3x+2}[/tex]

Thus, the answer is...

A?

Answer:

The right solution is option a : [tex]\frac{8x^2 - 18}{3x+2}[/tex]

Step-by-step explanation:

We have to simplify the following rational expression,

[tex]\frac{4x^3-9x}{2x-7}[/tex] ÷ [tex]\frac{3x^3+2x^2}{4x^2-14x}[/tex]

Take into account the rule (a / b) / (c / d) = ad / bc. This simplifies the expression a bit further --- (1)

[tex]\left(4x^3-9x\right)\left(4x^2-14x\right)[/tex] ÷ [tex]\left(2x-7\right)\left(3x^3+2x^2\right)[/tex]

Let's now factor each individual expression as demonstrated below. Afterwards we can substitute back into the expression above --- (2)

Goal - Factor : (4x³ - 9x)(4x² - 14x),

4x³ - 9x ⇒ x(4x² - 9)

4x² - 14x ⇒ 2x(2x - 7)

[tex]x\left(4x^2-9\right)\cdot \:2x\left(2x-7\right) = 2x^2\left(4x^2-9\right)\left(2x-7\right)[/tex]

That leaves us with the following expression,

[tex]2x^2\left(4x^2-9\right)\left(2x-7\right)[/tex] ÷ [tex]\left(2x-7\right)\left(3x^3+2x^2\right)[/tex]

As you can see the " 2x - 7 " cancel out, and the " x² " cancel as well, leaving us with a further simplified expression,

[tex]2\left(4x^2-9\right)[/tex] ÷ [tex]3x+2[/tex]

Which can also be rewritten as [tex]8x^2 - 18[/tex] ÷ [tex]3x + 2[/tex] = [tex]\frac{8x^2 - 18}{3x+2}[/tex], in other words the first option.

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