Simplify and state the domain for the expression.
(x^2 + 10x + 24) / (x^3 - x^2 - 20x)
[ (x+4)(x+6) ] / [ x(x-5)(x+4) ]
= (x+6) / [ x(x-5) ]
blows up if x = 0 or if x = 5
and has a hole if x = -4
thank you !!
To simplify the expression (x^2 + 10x + 24) / (x^3 - x^2 - 20x), let's first factor both the numerator and denominator:
Numerator:
x^2 + 10x + 24 factors to (x + 6)(x + 4).
Denominator:
x^3 - x^2 - 20x factors to x(x^2 - x - 20). The quadratic part, x^2 - x - 20, further factors to (x - 5)(x + 4).
Now, we can rewrite the expression as:
(x + 6)(x + 4) / x(x - 5)(x + 4)
To simplify further, we can cancel out the common factor of (x + 4) in both the numerator and denominator:
(x + 6)(x + 4) / x(x - 5)(x + 4) = (x + 6) / x(x - 5)
Now, let's determine the domain of the simplified expression. The expression will be undefined if the denominator is equal to zero. So, we need to find the values of x that make x(x - 5) equal to zero:
x(x - 5) = 0
This equation has two solutions: x = 0 and x = 5.
Therefore, the domain of the simplified expression is all real numbers except x = 0 and x = 5. In interval notation, the domain can be represented as (-∞, 0) U (0, 5) U (5, ∞).