Question

algebra: concepts and connections - plc
adding and subtracting rational expressions (continued)
henry correctly subtracted (\frac{3p + 5}{3p}) from (\frac{4p + 3}{p - 2}) and got (\frac{12p^2 + 9p - 3p^2 + p + 10}{(p - 2)(3p)}). what is the simplified difference?
options:
(\frac{9p^2 + 8p + 10}{(p - 2)(3p)})
(\frac{15p^2 + 10p + 10}{(p - 2)(3p)})
(\frac{9p^2 + 10p + 10}{(p - 2)(3p)})
(\frac{21p^2 - 2p + 10}{(p - 2)(3p)})

Explanation:

Step1: Define the subtraction

The problem is $\frac{4p+3}{p-2} - \frac{3p+5}{3p}$

Step2: Find common denominator

The common denominator is $(p-2)(3p)$. Rewrite each fraction:
$\frac{(4p+3)(3p)}{(p-2)(3p)} - \frac{(3p+5)(p-2)}{(p-2)(3p)}$

Step3: Expand numerators

First numerator: $(4p+3)(3p) = 12p^2 + 9p$
Second numerator: $(3p+5)(p-2) = 3p^2 -6p +5p -10 = 3p^2 -p -10$

Step4: Subtract numerators

$(12p^2 +9p) - (3p^2 -p -10) = 12p^2 +9p -3p^2 +p +10$
Simplify: $9p^2 +10p +10$

Step5: Combine into single fraction

$\frac{9p^2 +10p +10}{(p-2)(3p)}$

Answer:

$\frac{9p^2 +10p +10}{(p-2)(3p)}$