Question

guided practice
write the factors of each denominator.
2p + 10 \t 2(p + 5)
p² − 25 \t (p + 5)(p − 5)
combining rational expressions
step 2
rewrite the rational expressions with like denominator.
type the numerator in the left blank and the denominator in the right blank.
then click or tap done.
what are the missing factors?
\frac{3p}{2(p + 5)} \cdot \frac{\sqrt{}p - 5}{\sqrt{}} + \frac{2}{(p + 5)(p - 5)} \cdot \frac{\sqrt{}2}{\sqrt{}2}

Explanation:

Step1: Analyze first fraction

The original denominator of the first fraction is $2(p+5)$, and the target denominator is $(p+5)(p-5)$. So the missing factor is $\frac{(p+5)(p-5)}{2(p+5)}=\frac{p-5}{2}$? No, wait: the first fraction is $\frac{3p}{2(p+5)}$, to get denominator $(p+5)(p-5)$, multiply numerator and denominator by $(p-5)$. So the numerator becomes $3p(p-5)$, so the missing factor in the first blank is $p-5$.

Step2: Analyze second fraction

The original denominator of the second fraction is $p^2 - 25=(p+5)(p-5)$, and the target denominator is also $(p+5)(p-5)$. Wait, no, the second term is $\frac{2}{(p+5)(p-5)}$, wait no: the second original expression's denominator is $p^2 -25=(p+5)(p-5)$, so to make it have the same denominator as the first transformed fraction (which is $(p+5)(p-5)$), but wait the first term after multiplying by $(p-5)$ has denominator $2(p+5)(p-5)$? Wait no, looking at the given structure: the first blank is in the numerator of the first fraction's transformation, so $\frac{3p}{2(p+5)} \times \frac{p-5}{p-5} = \frac{3p(p-5)}{2(p+5)(p-5)}$, so the first missing factor is $p-5$. The second term: the original is $\frac{2}{(p+5)(p-5)}$, to get denominator $2(p+5)(p-5)$, multiply numerator and denominator by $2$, so the numerator becomes $2 \times 2 = 4$, so the missing factor in the second blank is $2$.

So the missing factors are $p-5$ and $2$.

Answer:

$p-5$; $2$