Find all numbers that must be excluded from the domain of the given rational expression.
1) x^2+4x/x^2-16
2) 6x^3y-6xy/x^3+5x^2+4x
3) 4-x^2/x^2+x-6
look at the denominators.
a. (x-4)(x+4)
b. x(x+5)(x+1)
c. (x+3)(x-2)
now I ask you: what values of x in each of these will make the product zero?
To find the numbers that must be excluded from the domain of a rational expression, we need to look for values of the variable that would make the denominator equal to zero. Dividing by zero is undefined in mathematics, so these values should be excluded.
1) x^2 + 4x / x^2 - 16
To find the excluded values, we set the denominator equal to zero and solve for x:
x^2 - 16 = 0
(x - 4)(x + 4) = 0
This equation has two solutions: x = 4 and x = -4. Therefore, the numbers 4 and -4 must be excluded from the domain.
2) 6x^3y - 6xy / x^3 + 5x^2 + 4x
We set the denominator equal to zero and solve for x:
x^3 + 5x^2 + 4x = 0
To find the solutions, we can factor out an x:
x(x^2 + 5x + 4) = 0
Then, we factor the quadratic expression:
(x + 1)(x + 4) = 0
This gives us two solutions: x = -1 and x = -4. Therefore, the numbers -1 and -4 must be excluded from the domain.
3) 4 - x^2 / x^2 + x - 6
Setting the denominator equal to zero and solving for x:
x^2 + x - 6 = 0
We can factor the quadratic expression:
(x + 3)(x - 2) = 0
This gives us two solutions: x = -3 and x = 2. Therefore, the numbers -3 and 2 must be excluded from the domain.
To find the numbers that must be excluded from the domain of a rational expression, we need to look for values of the variable that would make the denominator equal to zero. Division by zero is undefined, so these values need to be excluded from the domain.
1) For the first rational expression, x^2 - 16 is the denominator. We need to find the values of x that would make it equal to zero by setting it equal to zero and solving:
x^2 - 16 = 0
Factoring the equation gives us:
(x + 4)(x - 4) = 0
Setting each factor equal to zero gives us:
x + 4 = 0 or x - 4 = 0
Solving for x gives us:
x = -4 or x = 4
Therefore, the numbers -4 and 4 must be excluded from the domain.
2) For the second rational expression, x^3 + 5x^2 + 4x is the denominator. Setting it equal to zero and solving:
x^3 + 5x^2 + 4x = 0
Unfortunately, this equation cannot be easily factored, so we need to use methods like factoring by grouping or the rational root theorem to find the values of x that would make it equal to zero. Solving this equation may require more advanced techniques that cannot be explained in a simple text format.
3) For the third rational expression, x^2 + x - 6 is the denominator. We set it equal to zero and solve:
x^2 + x - 6 = 0
Factoring the equation gives us:
(x + 3)(x - 2) = 0
Setting each factor equal to zero gives us:
x + 3 = 0 or x - 2 = 0
Solving for x gives us:
x = -3 or x = 2
Therefore, the numbers -3 and 2 must be excluded from the domain.
In summary, for the given rational expressions:
1) The numbers -4 and 4 must be excluded from the domain.
2) Further calculations are needed to determine which numbers must be excluded from the domain.
3) The numbers -3 and 2 must be excluded from the domain.