Mathematics, 02.02.2020 15:51 KIAZTUNEZ

Simplify the function f(x)=1/3(81)^3x/4 . then determine the key aspects of the function. the initial value is . the simplified base is . the domain is . the range is .

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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

Polynomial Roots Calculator :

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

P Q P/Q F(P/Q) Divisor -1 1 -1.00 -82.00 -3 1 -3.00 -108.00 -9 1 -9.00 -810.00 -27 1 -27.00 -19764.00 -81 1 -81.00 -531522.00 1 1 1.00 -80.00 3 1 3.00 -54.00 9 1 9.00 648.00 27 1 27.00 19602.00 81 1 81.00 531360.00

Polynomial Roots Calculator found no rational roots

Equation at the end of step 1 : x3 - 81 = 0 Step 2 :Solving a Single Variable Equation :

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

One solution was found : x = 3 • ∛3 = 4.3267

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Simplify the function f(x)=1/3(81)^3x/4 . then determine the key aspects of the function. the initia...