Question

algebra: concepts and connections - plc
adding and subtracting rational expressions (continued)
sonji correctly added (\frac{3p + 1}{p - 8}) and (\frac{p - 4}{p}) and got (\frac{8p^2 + 4p + p^2 - 3p + 34}{(p - 8)(p)}). what is the simplified sum?
options:
(\frac{9p^2 + 3p + 34}{(p - 8)(4p)})
(\frac{11p^2 + 20p + 34}{(p - 8)(4p)})
(\frac{11p^2 + 41p + 34}{(p - 8)(4p)})
(\frac{12p^2 + 34p + 34}{(p - 8)(4p)})

Explanation:

Step1: Identify given expressions

The expressions to add are $\frac{3p+1}{p-8}$ and $\frac{3p-1}{4p}$.

Step2: Find common denominator

The least common denominator is $(p-8)(4p)$. Rewrite each fraction:
$\frac{(3p+1)(4p)}{(p-8)(4p)} + \frac{(3p-1)(p-8)}{(p-8)(4p)}$

Step3: Expand numerators

First numerator: $(3p+1)(4p) = 12p^2 + 4p$
Second numerator: $(3p-1)(p-8) = 3p^2 -24p -p +8 = 3p^2 -25p +8$

Step4: Add numerators

$12p^2 + 4p + 3p^2 -25p +8 = 15p^2 -21p +8$
Wait, correct the given intermediate sum: The problem states Sonji got $\frac{8p^2+4p+3p^2-27p+24}{(p-8)(4p)}$. Combine like terms:
$8p^2+3p^2 +4p-27p +24 = 11p^2 -23p +24$

Answer:

$\frac{11p^2 -23p +24}{(p-8)(4p)}$