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Solve for [tex]\( x \)[/tex].

[tex]\[ 3x = 6x - 2 \][/tex]


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TIME REMAINII
The volume of a rectangular prism is [tex]$\left(x^4+4 x^3+3 x^2+8 x+4\right)$[/tex], and the area of its base is [tex]$\left(x^3+3 x^2+8\right)$[/tex]. If the volum of a rectangular prism is the product of its base area and height, what is the height of the prism?
[tex]$x+1-\frac{4}{x^4+4 x^3+3 x^2+8 x+4}$[/tex]
[tex]$x+1+\frac{4}{x^4+4 x^3+3 x^2+8 x+4}$[/tex]
[tex]$x+1-\frac{4}{x^3+3 x^2+8}$[/tex]
[tex]$x+1+\frac{4}{x^3+3 x^2+8}$[/tex]
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Response:

The volume of a rectangular prism is [tex]\( x^4+4x^3+3x^2+8x+4 \)[/tex], and the area of its base is [tex]\( x^3+3x^2+8 \)[/tex].

If the volume of a rectangular prism is the product of its base area and height, what is the height of the prism?

A. [tex]\( x+1-\frac{4}{x^4+4x^3+3x^2+8x+4} \)[/tex]

B. [tex]\( x+1+\frac{4}{x^4+4x^3+3x^2+8x+4} \)[/tex]

C. [tex]\( x+1-\frac{4}{x^3+3x^2+8} \)[/tex]

D. [tex]\( x+1+\frac{4}{x^3+3x^2+8} \)[/tex]



Answer :

GinnyAnswer
To solve the problem of finding the height of the rectangular prism, we start by applying the formula for the volume of a rectangular prism, which is given by the product of its base area and height:

[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]

Given:
- The volume of the rectangular prism is [tex]\( V = x^4 + 4x^3 + 3x^2 + 8x + 4 \)[/tex]
- The area of the base of the rectangular prism is [tex]\( A = x^3 + 3x^2 + 8 \)[/tex]

We need to determine the height [tex]\( h \)[/tex] of the prism. We can derive the height by rearranging the volume formula to solve for [tex]\( h \)[/tex]:

[tex]\[ h = \frac{V}{A} \][/tex]

Substitute the expressions for the volume and base area:

[tex]\[ h = \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8} \][/tex]

Next, we simplify the expression [tex]\( \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8} \)[/tex]:

To simplify this rational function, consider polynomial long division:

1. The leading term of the numerator [tex]\( x^4 \)[/tex] divided by the leading term of the denominator [tex]\( x^3 \)[/tex]:
[tex]\[ x \][/tex]

2. Multiply the entire divisor [tex]\( x^3 + 3x^2 + 8 \)[/tex] by this quotient:
[tex]\[ x(x^3 + 3x^2 + 8) = x^4 + 3x^3 + 8x \][/tex]

3. Subtract this product from the numerator:
[tex]\[ (x^4 + 4x^3 + 3x^2 + 8x + 4) - (x^4 + 3x^3 + 8x) = x^4 + 4x^3 + 3x^2 + 8x + 4 - x^4 - 3x^3 - 8x = x^3 + 3x^2 + 4 \][/tex]

4. The new expression becomes:
[tex]\[ \frac{x^3 + 3x^2 + 4}{x^3 + 3x^2 + 8} \][/tex]

We notice that by performing polynomial division, we have:
[tex]\[ x^3 + 3x^2 + 4 \][/tex]

This shows a quotient simplification leading to the final result, combined with the original leading term quotient [tex]\( x \)[/tex]:

Now we test against the options to match it properly.

Upon simplifying and matching with the given options:
- [tex]\( x+1-\frac{4}{x^3+3 x^2+8} \)[/tex]
- [tex]\( x+1+\frac{4}{x^3+3 x^2+8} \)[/tex]
- [tex]\( x+1-\frac{4}{x^4+4 x^3+3 x^2+8 x+4} \)[/tex]
- [tex]\( x+1+\frac{4}{x^4+4 x^3+3 x^2+8 x+4} \)[/tex]

We find that:

[tex]\[ \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8} = x + 1 - \frac{4}{x^3 + 3x^2 + 8} \][/tex]

Thus, the height of the prism is:
[tex]\[ \boxed{3} \][/tex]

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