Question

algebra: concepts and connections - plc
adding and subtracting rational expressions (continued)
which statements are true about finding the difference of the expressions? choose three correct answers.
\\(\frac{3p + 1}{6p} - \frac{2p - 3}{2p^2}\\)
the first fraction rewritten with the lcd is \\(\frac{3p^2 + p}{6p^2}\\).
the common denominator is \\(6p\\).
the difference is a rational expression.

Explanation:

Step1: Find the LCD of denominators

Denominators are $6p=2\cdot3\cdot p$ and $2p^2=2\cdot p^2$. The least common denominator (LCD) is $2\cdot3\cdot p^2=6p^2$.

Step2: Rewrite first fraction with LCD

Multiply numerator and denominator of $\frac{3p+1}{6p}$ by $p$:
$\frac{(3p+1)\cdot p}{6p\cdot p}=\frac{3p^2+p}{6p^2}$

Step3: Analyze the difference result

Subtract the rewritten fractions:
$\frac{3p^2+p}{6p^2}-\frac{2p-3}{2p^2}=\frac{3p^2+p - 3(2p-3)}{6p^2}=\frac{3p^2+p-6p+9}{6p^2}=\frac{3p^2-5p+9}{6p^2}$, which is a rational expression.

Step4: Evaluate each statement

1. "The first fraction rewritten with the LCD is $\frac{3p^2+p}{6p^2}$": True (matches Step2).
2. "The common denominator is $6p$": False (LCD is $6p^2$).
3. "The difference is a rational expression": True (matches Step3 result, a ratio of polynomials).

Answer:

The three correct statements are:

1. The first fraction rewritten with the LCD is $\frac{3p^2+p}{6p^2}$.
2. The difference is a rational expression.

(Note: Since only three options are provided, the third correct statement would be confirmed by eliminating the false one, so the two true statements above plus the verification that the remaining claim is false identifies the valid selections.)