Step by Step Solution
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
88-(x^3)=0
Step by step solution :
Step 1 :
Trying to factor as a Difference of Cubes:
1.1 Factoring: 88-x3
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 : 88 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+88
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 88 and the Trailing Constant is -1.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8 ,11 ,22 ,44 ,88
of the Trailing Constant : 1
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 89.00 | ||||||
| -1 | 2 | -0.50 | 88.12 | ||||||
| -1 | 4 | -0.25 | 88.02 | ||||||
| -1 | 8 | -0.12 | 88.00 | ||||||
| -1 | 11 | -0.09 | 88.00 | ||||||
| -1 | 22 | -0.05 | 88.00 | ||||||
| -1 | 44 | -0.02 | 88.00 | ||||||
| -1 | 88 | -0.01 | 88.00 | ||||||
| 1 | 1 | 1.00 | 87.00 | ||||||
| 1 | 2 | 0.50 | 87.88 | ||||||
| 1 | 4 | 0.25 | 87.98 | ||||||
| 1 | 8 | 0.12 | 88.00 | ||||||
| 1 | 11 | 0.09 | 88.00 | ||||||
| 1 | 22 | 0.05 | 88.00 | ||||||
| 1 | 44 | 0.02 | 88.00 | ||||||
| 1 | 88 | 0.01 | 88.00 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 1 :
88 - x3 = 0
Step 2 :
Solving a Single Variable Equation :
2.1 Solve : -x3+88 = 0
Subtract 88 from both sides of the equation :
-x3 = -88
Multiply both sides of the equation by (-1) : x3 = 88
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:
x = ∛ 88
Can ∛ 88 be simplified ?
Yes! The prime factorization of 88 is
2•2•2•11
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).
∛ 88 = ∛ 2•2•2•11 =
2 • ∛ 11
The equation has one real solution
This solution is x = 2 • ∛11 = 4.4480
One solution was found :
x = 2 • ∛11 = 4.4480How did we do?
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