Solution - Reducing fractions to their lowest terms
Step by Step Solution
Step 1 :
4
Simplify ——
x2
Equation at the end of step 1 :
d 4
—— • (3x + ——)
dx x2
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Adding a fraction to a whole
Rewrite the whole as a fraction using x2 as the denominator :
3x 3x • x2
3x = —— = ———————
1 x2
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:
3x • x2 + 4 3x3 + 4
——————————— = ———————
x2 x2
Equation at the end of step 2 :
d (3x3 + 4)
—— • —————————
dx x2
Step 3 :
d
Simplify ——
dx
Canceling Out :
3.1 Canceling out d as it appears on both sides of the fraction line
Equation at the end of step 3 :
1 (3x3 + 4)
— • —————————
x x2
Step 4 :
Trying to factor as a Sum of Cubes :
4.1 Factoring: 3x3+4
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 : 3 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
4.2 Find roots (zeroes) of : F(x) = 3x3+4
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 3 and the Trailing Constant is 4.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,4
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 1.00 | ||||||
| -1 | 3 | -0.33 | 3.89 | ||||||
| -2 | 1 | -2.00 | -20.00 | ||||||
| -2 | 3 | -0.67 | 3.11 | ||||||
| -4 | 1 | -4.00 | -188.00 | ||||||
| -4 | 3 | -1.33 | -3.11 | ||||||
| 1 | 1 | 1.00 | 7.00 | ||||||
| 1 | 3 | 0.33 | 4.11 | ||||||
| 2 | 1 | 2.00 | 28.00 | ||||||
| 2 | 3 | 0.67 | 4.89 | ||||||
| 4 | 1 | 4.00 | 196.00 | ||||||
| 4 | 3 | 1.33 | 11.11 |
Polynomial Roots Calculator found no rational roots
Multiplying exponential expressions :
4.3 x1 multiplied by x2 = x(1 + 2) = x3
Trying to factor as a Sum of Cubes :
4.4 Factoring: 3x3+4
Check : 3 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Final result :
3x3 + 4
———————
x3
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