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: x3-42
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 : 42 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) = x3-42
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 -42.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,6 ,7 ,14 ,21 ,42
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -43.00 | ||||||
| -2 | 1 | -2.00 | -50.00 | ||||||
| -3 | 1 | -3.00 | -69.00 | ||||||
| -6 | 1 | -6.00 | -258.00 | ||||||
| -7 | 1 | -7.00 | -385.00 | ||||||
| -14 | 1 | -14.00 | -2786.00 | ||||||
| -21 | 1 | -21.00 | -9303.00 | ||||||
| -42 | 1 | -42.00 | -74130.00 | ||||||
| 1 | 1 | 1.00 | -41.00 | ||||||
| 2 | 1 | 2.00 | -34.00 | ||||||
| 3 | 1 | 3.00 | -15.00 | ||||||
| 6 | 1 | 6.00 | 174.00 | ||||||
| 7 | 1 | 7.00 | 301.00 | ||||||
| 14 | 1 | 14.00 | 2702.00 | ||||||
| 21 | 1 | 21.00 | 9219.00 | ||||||
| 42 | 1 | 42.00 | 74046.00 |
Polynomial Roots Calculator found no rational roots
Final result :
x3 - 42
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