Solution - Factoring binomials using the difference of squares
Other Ways to Solve
Factoring binomials using the difference of squaresStep by Step Solution
Step 1 :
Equation at the end of step 1 :
(((n2)+2)1)
——————————— ÷ ((2n3+15)1) ÷ 3
3
Step 2 :
n2 + 2
Simplify ——————
3
Polynomial Roots Calculator :
2.1 Find roots (zeroes) of : F(n) = n2 + 2
Polynomial Roots Calculator is a set of methods aimed at finding values of n for which F(n)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers n 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 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 | 3.00 | ||||||
| -2 | 1 | -2.00 | 6.00 | ||||||
| 1 | 1 | 1.00 | 3.00 | ||||||
| 2 | 1 | 2.00 | 6.00 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 2 :
(n2 + 2)
———————— ÷ (2n3 + 15) ÷ 3
3
Step 3 :
n2+2
Divide ———— by 2n3+15
3
Trying to factor as a Sum of Cubes :
3.1 Factoring: 2n3 + 15
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(n) = 2n3 + 15
See theory in step 2.1
In this case, the Leading Coefficient is 2 and the Trailing Constant is 15.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,3 ,5 ,15
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 13.00 | ||||||
| -1 | 2 | -0.50 | 14.75 | ||||||
| -3 | 1 | -3.00 | -39.00 | ||||||
| -3 | 2 | -1.50 | 8.25 | ||||||
| -5 | 1 | -5.00 | -235.00 | ||||||
| -5 | 2 | -2.50 | -16.25 | ||||||
| -15 | 1 | -15.00 | -6735.00 | ||||||
| -15 | 2 | -7.50 | -828.75 | ||||||
| 1 | 1 | 1.00 | 17.00 | ||||||
| 1 | 2 | 0.50 | 15.25 | ||||||
| 3 | 1 | 3.00 | 69.00 | ||||||
| 3 | 2 | 1.50 | 21.75 | ||||||
| 5 | 1 | 5.00 | 265.00 | ||||||
| 5 | 2 | 2.50 | 46.25 | ||||||
| 15 | 1 | 15.00 | 6765.00 | ||||||
| 15 | 2 | 7.50 | 858.75 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 3 :
(n2 + 2)
—————————————— ÷ 3
3 • (2n3 + 15)
Step 4 :
n2+2
Divide —————————— by 3
3•(2n3+15)
Trying to factor as a Sum of Cubes :
4.1 Factoring: 2n3 + 15
Check : 2 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Trying to factor as a Sum of Cubes :
4.2 Factoring: 2n3 + 15
Check : 2 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Final result :
n2 + 2
——————————————
9 • (2n3 + 15)
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