Respuesta :
x3-81=0 One solution was found : x = 3 • ∛3 = 4.3267Step by step solution :Step 1 :Trying to factor as a Difference of Cubes:
1.1 Factoring: x3-81
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0+b3 =
a3+b3
Check : 81 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
1.2 Find roots (zeroes) of : F(x) = x3-81
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is -81.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,9 ,27 ,81
Let us test ....
Polynomial Roots Calculator found no rational roots
2.1 Solve : x3-81 = 0
Add 81 to both sides of the equation :
x3 = 81
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:
x = ∛ 81
Can ∛ 81 be simplified ?
Yes! The prime factorization of 81 is
3•3•3•3
To be able to remove something from under the radical, there have to be 3 instances of it (because we are taking a cube i.e. cube root).
∛ 81 = ∛ 3•3•3•3 =
3 • ∛ 3
The equation has one real solution
This solution is x = 3 • ∛3 = 4.3267
Answer:
The initial value is
✔ 1/3
The simplified base is
✔ 27
The domain is
✔ all real numbers
The range is
✔ y > 0
Step-by-step explanation:
