Simplify
1... x+5/x^2+6x+5
Answer... 1/x+1 where x is not equal to -1, -5 (Is this correct? Or is it just -1? Or just 5?)
2... x^2+3x-4/x+4
Answer... x-1 where x is not equal to 1 (or -4...?)
since the demominator is (x+1)(x+5) both x ≠ both -1 and -5
when x = -5, you have 0/0, which is also undefined.
for the 2nd problem. whenever the denominator is zero, division is undefined, so x ≠ -4
when x = 1, you just have 0/5 = 0, which is perfectly fine
Thx
1. To simplify the expression x+5/x^2+6x+5, we can begin by factoring the denominator:
x^2 + 6x + 5 = (x + 1)(x + 5)
Now, we can rewrite the expression as:
(x + 5)/(x + 1)(x + 5)
Next, we can cancel out the (x + 5) term in the numerator and denominator, leaving us with:
1/(x + 1)
Therefore, the simplified expression is 1/(x + 1), where x is not equal to -1.
In this case, we only exclude -1 as a possible value for x, since x + 1 is in the denominator.
2. To simplify the expression x^2+3x-4/x+4, we can first factor the numerator:
x^2 + 3x - 4 = (x - 1)(x + 4)
Now, we can rewrite the expression as:
(x - 1)(x + 4)/(x + 4)
Next, we can cancel out the (x + 4) term in the numerator and denominator, leaving us with:
(x - 1)
Therefore, the simplified expression is x - 1, where x is not equal to -4.
In this case, we exclude both x = -4 and x = 1 as possible values for x because they would make the denominator equal to zero, which is undefined.
To simplify the expressions, you need to factorize the numerator and denominator separately and then cancel out any common factors if possible.
1. Simplifying (x + 5) / (x^2 + 6x + 5):
The numerator (x + 5) and denominator (x^2 + 6x + 5) can both be factored.
The numerator is already in its simplest form.
The denominator can be factored as (x + 1)(x + 5).
Therefore, (x + 5) / (x^2 + 6x + 5) simplifies to 1 / (x + 1)(x + 5).
Note that the value of x cannot be equal to -1 or -5 because it would make the denominator equal to zero, and division by zero is undefined.
2. Simplifying (x^2 + 3x - 4) / (x + 4):
The numerator (x^2 + 3x - 4) and denominator (x + 4) can both be factored.
The numerator can be factored as (x + 4)(x - 1).
The denominator is already in its simplest form.
Therefore, (x^2 + 3x - 4) / (x + 4) simplifies to (x + 4)(x - 1) / (x + 4).
However, before canceling out the common factor, we need to ensure that this cancellation is valid. In other words, we need to ensure that x = -4 is not part of the solution.
By substituting x = -4 into the original expression, we get (-4)^2 + 3(-4) - 4 / (-4 + 4) = 0 / 0, which is undefined. Therefore, x = -4 is not allowed.
After excluding x = -4, the expression can be further simplified by canceling out the common factor of (x + 4):
(x + 4)(x - 1) / (x + 4) simplifies to x - 1.
Note that the value of x cannot be equal to 1 because it would make the denominator equal to zero, and division by zero is undefined.