Question

Write the rational expression in lowest terms.
StartFraction hj minus hp minus 2 j plus 2 p Over hj minus hp plus 2 j minus 2 p EndFraction

Answer 1

To simplify the rational expression

\[
\frac{hj - hp - 2j + 2p}{hj - hp + 2j - 2p}
\]

we can start by rearranging the terms in both the numerator and the denominator.

**Numerator:**
\[
hj - hp - 2j + 2p \quad \text{can be rearranged as} \quad hj - hp + 2p - 2j
\]

**Denominator:**
\[
hj - hp + 2j - 2p \quad \text{can also be rearranged as} \quad hj - hp + 2j - 2p
\]

Now let's factor both the numerator and the denominator.

1. **Factoring the numerator:**
\[
hj - hp + 2p - 2j = h(j - p) + 2(p - j)
\]
Notice that \(2(p - j) = -2(j - p)\), so we can group terms:
\[
= h(j - p) - 2(j - p) = (j - p)(h - 2)
\]

2. **Factoring the denominator:**
\[
hj - hp + 2j - 2p = h(j - p) + 2(j - p) = (j - p)(h + 2)
\]

Now substituting back into the original rational expression:

\[
\frac{(j - p)(h - 2)}{(j - p)(h + 2)}
\]

Assuming \(j \neq p\) (to avoid division by zero), we can cancel out the common factor \((j - p)\):

\[
\frac{h - 2}{h + 2}
\]

Thus, the rational expression in lowest terms is:

\[
\frac{h - 2}{h + 2}
\]

This is the final simplified form.