evaluate : (x^3 + 2/x^3) (x - 2/x)
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Reducing fractions to their lowest terms
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We think you wrote:
(x^3+2/x^3)(x-2/x)
This deals with reducing fractions to their lowest terms.
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1 solution(s) found
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Step by Step Solution
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STEP
1
:
2
Simplify —
x
Equation at the end of step
1
:
2 2
((x3)+————)•(x-—)
(x3) x
STEP
2
:
Rewriting the whole as an Equivalent Fraction
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using x as the denominator :
x x • x
x = — = —————
1 x
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
2.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
x • x - (2) x2 - 2
——————————— = ——————
x x
Equation at the end of step
2
:
2 (x2 - 2)
((x3) + ————) • ————————
(x3) x
STEP
3
:
2
Simplify ——
x3
Equation at the end of step
3
:
2 (x2 - 2)
((x3) + ——) • ————————
x3 x
STEP
4
:
Rewriting the whole as an Equivalent Fraction :
4.1 Adding a fraction to a whole
Rewrite the whole as a fraction using x3 as the denominator :
x3 x3 • x3
x3 = —— = ———————
1 x3
Adding fractions that have a common denominator :
4.2 Adding up the two equivalent fractions
x3 • x3 + 2 x6 + 2
——————————— = ——————
x3 x3
Equation at the end of step
4
:
(x6 + 2) (x2 - 2)
———————— • ————————
x3 x
STEP
5
:
Trying to factor as a Sum of Cubes:
5.1 Factoring: x6+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 :
5.2 Find roots (zeroes) of : F(x) = x6+2
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 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 66.00
1 1 1.00 3.00
2 1 2.00 66.00
Polynomial Roots Calculator found no rational roots
Trying to factor as a Difference of Squares:
5.3 Factoring: x2-2
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 2 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares.
Multiplying exponential expressions :
5.4 x3 multiplied by x1 = x(3 + 1) = x4
Trying to factor as a Sum of Cubes:
5.5 Factoring: x6+2
Check : 2 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Final result :
(x6 + 2) • (x2 + 2)
———————————————————
x4
Terms and topics
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