Simplify each expression and state any non-permissible values. (4 marks - show your work)
x^2-2x/x+1 x x^2-1/x^2+x-6
To simplify the first expression, we can factor the numerator and denominator if possible and then cancel out any common factors.
Expression 1: (x^2 - 2x) / (x + 1)
Step 1: Factor the numerator (if possible).
The numerator is already in factored form.
Step 2: Factor the denominator (if possible).
The denominator (x + 1) is already in factored form.
Step 3: Cancel out any common factors.
Since there are no common factors that can be canceled out, the expression cannot be simplified further.
Non-permissible values:
To find the non-permissible values, we need to identify any values of x that would make the denominator equal to zero, as division by zero is undefined.
In this case, the denominator is (x + 1), so the non-permissible value is when x = -1.
Now, let's move on to simplifying the second expression.
Expression 2: (x^2 - 1) / (x^2 + x - 6)
Step 1: Factor the numerator (if possible).
The numerator (x^2 - 1) can be factored using the difference of squares formula:
(x^2 - 1) = (x - 1)(x + 1)
Step 2: Factor the denominator (if possible).
The denominator (x^2 + x - 6) can be factored by finding two numbers that multiply to -6 and add up to +1:
(x^2 + x - 6) = (x + 3)(x - 2)
Step 3: Cancel out any common factors.
Now, we can cancel out any common factors between the numerator and denominator.
(x + 1) and (x - 1) from the numerator can be canceled out with the same factors from the denominator.
Simplified expression: (x - 1) / (x + 3)
Non-permissible values:
The non-permissible values are the ones that make the denominator equal to zero: x = -3.
Therefore, the simplified expressions and their non-permissible values are:
Expression 1: (x^2 - 2x) / (x + 1) with a non-permissible value of x = -1.
Expression 2: (x - 1) / (x + 3) with a non-permissible value of x = -3.