Question

express in simplest form 3x+18\4x+8 divided by x^2-36/x+2

Answer 1

To simplify the expression \((\frac{3x + 18}{4x + 8}) \div (\frac{x^2 - 36}{x + 2})\), we can follow these steps:

1. **Rewrite the division as multiplication by the reciprocal**:

\[
\frac{3x + 18}{4x + 8} \cdot \frac{x + 2}{x^2 - 36}
\]

2. **Factor all the polynomials involved**:

- The numerator \(3x + 18\) can be factored as:

\[
3(x + 6)
\]

- The denominator \(4x + 8\) can be factored as:

\[
4(x + 2)
\]

- The expression \(x^2 - 36\) is a difference of squares and can be factored as:

\[
(x - 6)(x + 6)
\]

Now substituting these factorizations into the expression, we have:

\[
\frac{3(x + 6)}{4(x + 2)} \cdot \frac{x + 2}{(x - 6)(x + 6)}
\]

3. **Now multiply the fractions**:

This becomes:

\[
\frac{3(x + 6)(x + 2)}{4(x + 2)(x - 6)(x + 6)}
\]

4. **Cancel common factors**:

The \(x + 2\) and \(x + 6\) terms in the numerator and denominator can be canceled out:

\[
\frac{3}{4(x - 6)}
\]

So the simplified form of the given expression is:

\[
\frac{3}{4(x - 6)}
\]

This is the simplest form of the expression.