Solution - Factoring binomials using the difference of squares
Step by Step Solution
Step 1 :
d
Simplify ——
dx
Canceling Out :
1.1 Canceling out d as it appears on both sides of the fraction line
Equation at the end of step 1 :
1
(— • (x2 + 5)) • (x3 + 2)
x
Step 2 :
Polynomial Roots Calculator :
2.1 Find roots (zeroes) of : F(x) = x2+5
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 5.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,5
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 6.00 | ||||||
| -5 | 1 | -5.00 | 30.00 | ||||||
| 1 | 1 | 1.00 | 6.00 | ||||||
| 5 | 1 | 5.00 | 30.00 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 2 :
(x2 + 5)
———————— • (x3 + 2)
x
Step 3 :
Trying to factor as a Sum of Cubes :
3.1 Factoring: x3+2
Theory : A sum 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 : 2 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
3.2 Find roots (zeroes) of : F(x) = x3+2
See theory in step 2.1
In this case, the Leading Coefficient is 1 and the Trailing Constant is 2.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 1.00 | ||||||
| -2 | 1 | -2.00 | -6.00 | ||||||
| 1 | 1 | 1.00 | 3.00 | ||||||
| 2 | 1 | 2.00 | 10.00 |
Polynomial Roots Calculator found no rational roots
Trying to factor as a Sum of Cubes :
3.3 Factoring: x3+2
Check : 2 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Final result :
(x2 + 5) • (x3 + 2)
———————————————————
x
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