Question

adding and subtracting rational expressions: practice

\\(l = 2x\\)
\\(w = x - 1\\)
\\(h = x + 2\\)

which expressions represent the difference of the ratio of the length to the surface area and the ratio of the width to the surface area?
recall that \\(\text{surface area} = 2lw + 2lh + 2wh\\), where \\(l\\) is length, \\(w\\) is width, and \\(h\\) is height.

\\(\frac{2x}{10x^2 + 6x - 4} + \frac{x - 1}{10x^2 + 6x - 4}\\)
\\(\frac{3x - 1}{4(5x^2 - 2)(x + 1)}\\)
\\(\frac{1}{2(5x - 2)}\\)
\\(\frac{3x - 1}{2(x^3 + x^2 - 2x)}\\)
\\(\frac{2x}{10x^2 + 6x - 4} - \frac{x - 1}{10x^2 + 6x - 4}\\)

Explanation:

Calculate the surface area expression

Using the Polynomial Operations knowledge point

$$
LATEXBLOCK0
$$

Set up the difference of ratios

Using the Rational Expressions knowledge point

$$
LATEXBLOCK1
$$

Factor the surface area expression

Using the Polynomial Operations knowledge point

$$
LATEXBLOCK2
$$

Simplify the difference expression

Using the Simplifying Rational Expressions knowledge point

$$
LATEXBLOCK3
$$

Identify the matching choices

We compare our derived expressions with the given options:

1. \(\frac{2x}{10x^2 + 6x - 4} - \frac{x-1}{10x^2 + 6x - 4}\) matches the fifth option.
2. \(\frac{1}{2(5x-2)}\) matches the third option.

Answer:

• (A) \(\frac{2x}{10x^2 + 6x - 4} + \frac{x-1}{10x^2 + 6x - 4}\)
• (B) \(\frac{3x-1}{4(5x^2-2)(x+1)}\)
• (C) \(\frac{1}{2(5x-2)}\) (Correct answer)
• (D) \(\frac{3x-1}{2(x^3+x^2-2x)}\)
• (E) \(\frac{2x}{10x^2 + 6x - 4} - \frac{x-1}{10x^2 + 6x - 4}\) (Correct answer)