Question(c) $\frac{25-36 n^{2}}{m^{2}+m-2} \div \frac{5+6 n}{m+2}$ .

Studdy Solution

STEP 1

1. The problem involves simplifying a complex fraction involving polynomials in $n$ and $m$ .
2. Polynomial division and simplification techniques will be necessary.
3. The division operation can be converted into a multiplication by the reciprocal.

STEP 2

1. Factor the numerator and denominator of the first fraction.
2. Factor the numerator and denominator of the second fraction.
3. Convert the division to multiplication by the reciprocal of the second fraction.
4. Simplify the resulting expression by canceling common factors.

STEP 3

Factor the numerator of the first fraction $\frac{25-36 n^{2}}{m^{2}+m-2}$ .
$25 - 36n^2 = (5 - 6n)(5 + 6n)$

STEP 4

Factor the denominator of the first fraction $\frac{25-36 n^{2}}{m^{2}+m-2}$ .
$m^2 + m - 2 = (m + 2)(m - 1)$

STEP 5

Factor the numerator of the second fraction $\frac{5+6 n}{m+2}$ .
The numerator $5 + 6n$ is already in its simplest form.

STEP 6

Factor the denominator of the second fraction $\frac{5+6 n}{m+2}$ .
The denominator $m + 2$ is already in its simplest form.

STEP 7

Convert the division to multiplication by the reciprocal of the second fraction.
$\frac{(5 - 6n)(5 + 6n)}{(m + 2)(m - 1)} \div \frac{5+6n}{m+2} = \frac{(5 - 6n)(5 + 6n)}{(m + 2)(m - 1)} \times \frac{m+2}{5+6n}$

STEP 8

Simplify the resulting expression by canceling common factors.
$\frac{(5 - 6n)(5 + 6n)}{(m + 2)(m - 1)} \times \frac{m+2}{5+6n} = \frac{(5 - 6n) \cancel{(5 + 6n)}}{\cancel{(m + 2)}(m - 1)} \times \frac{\cancel{m+2}}{\cancel{5+6n}} = \frac{5 - 6n}{m - 1}$
Solution: The simplified expression is $\frac{5 - 6n}{m - 1}$ .

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