Questions: r(x)=(x^2+6x-7)/(x^2+2x-3)

Solution Steps

To simplify the rational function \( r(x) = \frac{x^2 + 6x - 7}{x^2 + 2x - 3} \), we need to factor both the numerator and the denominator and then cancel out any common factors.

We start with the rational function

\[ r(x) = \frac{x^2 + 6x - 7}{x^2 + 2x - 3} \]

Factoring the numerator \(x^2 + 6x - 7\) gives us

\[ x^2 + 6x - 7 = (x - 1)(x + 7) \]

Factoring the denominator \(x^2 + 2x - 3\) results in

\[ x^2 + 2x - 3 = (x - 1)(x + 3) \]

Now substituting the factored forms back into the rational function, we have:

\[ r(x) = \frac{(x - 1)(x + 7)}{(x - 1)(x + 3)} \]

We can cancel the common factor \((x - 1)\) from the numerator and the denominator, provided \(x \neq 1\):

\[ r(x) = \frac{x + 7}{x + 3} \quad (x \neq 1) \]

Final Answer