Multiplying/Dividing Rational Expressions

Question

Evaluate x2−2x−3 / x-3. when x=−2

Answer 1

Normally, one would attempt to factor all expressions to simplify a rational expression. But in the case of a substitution one might be eliminating the value which makes the expression undefined, that is , a 0 denominator.

If you factor the numerator you get [(x-3)(x+1)]/(x-3). Since we are asked to find the value of the expression at x = -2, we can reduce the expression and obtain x+1. At x = -2 we have [(-2) + 1] = -1

If we were asked to evaluate at x = 3, we could not factor, reduce and get 4 because the expression is not defined at x = 3.

Answer 2

Evaluate (x² − 2x − 3) / (x − 3). when x = −2

The simplest way to do this is to just plug in −2 for x to get

((−2)² − 2(−2) − 3) / (−2 − 3)

then calculate the result.

The other way is to factor the numerator

x² − 2x − 3 = (x − 3)(x + 1)

and use this to simplify the rational expression

For all x ≠ 3

(x² − 2x − 3) / (x − 3) = (x − 3)(x + 1) / (x − 3) = x + 1

Evaluate x + 1 at x = –2 and you get the same answer plugging in −2 for x in the original form of the function.

I only mention factoring the numerator and simplifying because I think it's likely to come up as you cover rational expressions in your course.