Solution - Simplification or other simple results
Other Ways to Solve
Simplification or other simple resultsStep by Step Solution
Step 1 :
Trying to factor as a Difference of Cubes:
1.1 Factoring: x3-56
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 : 56 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) = x3-56
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 -56.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,7 ,8 ,14 ,28 ,56
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -57.00 | ||||||
| -2 | 1 | -2.00 | -64.00 | ||||||
| -4 | 1 | -4.00 | -120.00 | ||||||
| -7 | 1 | -7.00 | -399.00 | ||||||
| -8 | 1 | -8.00 | -568.00 | ||||||
| -14 | 1 | -14.00 | -2800.00 | ||||||
| -28 | 1 | -28.00 | -22008.00 | ||||||
| -56 | 1 | -56.00 | -175672.00 | ||||||
| 1 | 1 | 1.00 | -55.00 | ||||||
| 2 | 1 | 2.00 | -48.00 | ||||||
| 4 | 1 | 4.00 | 8.00 | ||||||
| 7 | 1 | 7.00 | 287.00 | ||||||
| 8 | 1 | 8.00 | 456.00 | ||||||
| 14 | 1 | 14.00 | 2688.00 | ||||||
| 28 | 1 | 28.00 | 21896.00 | ||||||
| 56 | 1 | 56.00 | 175560.00 |
Polynomial Roots Calculator found no rational roots
Final result :
x3 - 56
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