Question

add \\(\frac{4p^2 + 6}{4p^2} + \frac{2p - 2}{10p}\\). which statements are true about adding the rational expressions? choose three correct answers. the sum is a rational expression. the first fraction rewritten with the lcd is \\(\frac{20p^2 + 30}{20p^2}\\). the sum is \\(\frac{24p^2 - 4p + 30}{20p^2} = \frac{12p^2 - 2p + 15}{10p^2}\\).

Explanation:

Step1: Factor denominators

First denominator: $4p^2 = 2^2p^2$
Second denominator: $10p = 2 \cdot 5p$

Step2: Find LCD

LCD = $2^2 \cdot 5 \cdot p^2 = 20p^2$

Step3: Rewrite first fraction

Multiply numerator/denominator by 5:
$\frac{(4p^2+6) \cdot 5}{4p^2 \cdot 5} = \frac{20p^2+30}{20p^2}$

Step4: Rewrite second fraction

Multiply numerator/denominator by 2p:
$\frac{(2p-2) \cdot 2p}{10p \cdot 2p} = \frac{4p^2-4p}{20p^2}$

Step5: Add the fractions

$\frac{20p^2+30 + 4p^2-4p}{20p^2} = \frac{24p^2-4p+30}{20p^2}$
Simplify by dividing numerator/denominator by 2:
$\frac{12p^2-2p+15}{10p^2}$

Step6: Verify rationality

A sum of rational expressions is rational.

Answer:

1. The sum is a rational expression.
2. The first fraction rewritten with the LCD is $\frac{20p^2+30}{20p^2}$.
3. The sum is $\frac{24p^2-4p+30}{20p^2} = \frac{12p^2-2p+15}{10p^2}$.