Question

Write the rational expression with the indicated denominator.
StartFraction 14 m minus n Over m squared plus mn plus n squared EndFraction
equalsStartFraction question mark Over n cubed minus m cubed EndFraction

Answer 1

To rewrite the rational expression with the indicated denominator, we want to express

\[
\frac{14m - n}{m^2 + mn + n^2}
\]

as

\[
\frac{?}{n^3 - m^3}.
\]

First, we know that the denominator \( n^3 - m^3 \) can be factored using the difference of cubes formula:

\[
n^3 - m^3 = (n - m)(n^2 + nm + m^2).
\]

We see that the denominator in the original fraction, \( m^2 + mn + n^2 \), is actually one of the factors in the factored form of \( n^3 - m^3 \).

Thus, we can rewrite the expression:

\[
\frac{14m - n}{m^2 + mn + n^2} = \frac{14m - n}{(n - m)(n^2 + nm + m^2)}.
\]

To write this with the denominator of \( n^3 - m^3 \), we multiply both the numerator and denominator of our original expression by \( n - m \):

\[
\frac{(14m - n)(n - m)}{(m^2 + mn + n^2)(n - m)} = \frac{(14m - n)(n - m)}{n^3 - m^3}.
\]

Now, the numerator is \( (14m - n)(n - m) \). Therefore, the rewritten expression is:

\[
\frac{(14m - n)(n - m)}{n^3 - m^3}.
\]

Thus, the expression you're looking for, represented in terms of the desired denominator, is

\[
(14m - n)(n - m).
\]