To simplify the given algebraic expression [tex]\(\frac{x-5}{x^2-2x-3}\)[/tex]:
1. Factorize the denominator:
The expression in the denominator is [tex]\(x^2 - 2x - 3\)[/tex]. We need to factorize this quadratic expression.
To factorize [tex]\(x^2 - 2x - 3\)[/tex], we look for two numbers that multiply to [tex]\(-3\)[/tex] (the constant term) and add up to [tex]\(-2\)[/tex] (the coefficient of the linear term [tex]\(x\)[/tex]).
These two numbers are [tex]\( -3 \)[/tex] and [tex]\( 1\)[/tex], since:
[tex]\[
(-3) \times 1 = -3 \quad \text{and} \quad (-3) + 1 = -2
\][/tex]
Therefore, we can write the quadratic expression as:
[tex]\[
x^2 - 2x - 3 = (x - 3)(x + 1)
\][/tex]
2. Rewrite the original expression using the factorized form of the denominator:
[tex]\[
\frac{x-5}{x^2-2x-3} = \frac{x-5}{(x-3)(x+1)}
\][/tex]
3. Simplify the expression:
It's important to check if the numerator [tex]\((x-5)\)[/tex] has any common factor with the denominator factors [tex]\((x-3)\)[/tex] and [tex]\((x+1)\)[/tex]. In this case, there are no common factors, so the expression cannot be simplified further.
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{5 - x}{x^2 - 2x - 3}
\][/tex]
By using the factorized form of the denominator and simplifying if possible, we arrive at our final form.
