Simplify each expression and state any non-permissible values. (4 marks - show your work)
x^2-2x divided by x+1 multiplied by x^2-1 divided by x^2+x-6 is the expression
x(x-2) /(x+1) * (x+1)(x-1)/[(x+3)(x-2)]
= x (x-1)/(x+3)
I read that as:
(x^2 - 2x)/(x+1) * (x^2 - 1)/(x^2 + x - 6)
=x(x-2)/(x+1)*(x-1)(x+1)/( (x+3)(x-2)
= x(x-1)/(x+3) or (x^2 - x)/(x+3) , x ≠ -1,2
I think it just blows up if x = -3
The inclusive vs exclusive restrictions in rational expressions always leads to debate.
I restricted x = -1 and x = -2 since subbing in these values in the original yields 0/0 but subbing those values in my answer (and yours) yields a real number
Since I was using the equal sign, the original is not equal to the final expression for x = -1, 2
but I did not state that x ≠ 3 since by default
the expression blows up in both the original and the final answer.
Some texts will state as restrictions all that cause a zero at the bottom, while others use the limitations I stated.
To simplify the given expression, we'll start by factoring the numerator and denominator separately.
1. Factor the numerator, x^2 - 2x:
x^2 - 2x = x(x - 2)
2. Factor the first denominator, x + 1, which is already in its simplest form.
3. Factor the second denominator, x^2 - 1:
x^2 - 1 is a difference of squares, so we can write it as (x - 1)(x + 1).
Now, let's rewrite the expression with the factored numerator and denominators:
(x(x - 2))/(x + 1) * [(x - 1)(x + 1)]/(x^2 + x - 6)
Next, we can simplify the expression by canceling out common factors. In this case, (x + 1) cancels out in both the numerator and denominator, leaving:
(x(x - 2))/(x^2 + x - 6)
To find any non-permissible values, we need to consider the values of x that would make the denominator equal to zero, as division by zero is undefined. Thus, we need to solve the equation x^2 + x - 6 = 0.
Factoring this quadratic equation, we get:
(x - 2)(x + 3) = 0
Setting each factor equal to zero and solving for x, we find two non-permissible values:
x - 2 = 0 -> x = 2
x + 3 = 0 -> x = -3
Therefore, the non-permissible values for x are x = 2 and x = -3.