Solution - Reducing fractions to their lowest terms
Other Ways to Solve
Reducing fractions to their lowest termsStep by Step Solution
Step 1 :
45
Simplify ——
x2
Equation at the end of step 1 :
45
((((x2) - 4x) - ——) - x) - 30
x2
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using x2 as the denominator :
x2 - 4x (x2 - 4x) • x2
x2 - 4x = ——————— = ——————————————
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
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
x2 - 4x = x • (x - 4)
Adding fractions that have a common denominator :
3.2 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-4) • x2 - (45) x4 - 4x3 - 45
————————————————————— = —————————————
x2 x2
Equation at the end of step 3 :
(x4 - 4x3 - 45)
(——————————————— - x) - 30
x2
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x2 as the denominator :
x x • x2
x = — = ——————
1 x2
Polynomial Roots Calculator :
4.2 Find roots (zeroes) of : F(x) = x4 - 4x3 - 45
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 -45.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,5 ,9 ,15 ,45
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -40.00 | ||||||
| -3 | 1 | -3.00 | 144.00 | ||||||
| -5 | 1 | -5.00 | 1080.00 | ||||||
| -9 | 1 | -9.00 | 9432.00 | ||||||
| -15 | 1 | -15.00 | 64080.00 | ||||||
| -45 | 1 | -45.00 | 4465080.00 | ||||||
| 1 | 1 | 1.00 | -48.00 | ||||||
| 3 | 1 | 3.00 | -72.00 | ||||||
| 5 | 1 | 5.00 | 80.00 | ||||||
| 9 | 1 | 9.00 | 3600.00 | ||||||
| 15 | 1 | 15.00 | 37080.00 | ||||||
| 45 | 1 | 45.00 | 3736080.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
4.3 Adding up the two equivalent fractions
(x4-4x3-45) - (x • x2) x4 - 5x3 - 45
—————————————————————— = —————————————
x2 x2
Equation at the end of step 4 :
(x4 - 5x3 - 45)
——————————————— - 30
x2
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x2 as the denominator :
30 30 • x2
30 = —— = ———————
1 x2
Polynomial Roots Calculator :
5.2 Find roots (zeroes) of : F(x) = x4 - 5x3 - 45
See theory in step 4.2
In this case, the Leading Coefficient is 1 and the Trailing Constant is -45.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,5 ,9 ,15 ,45
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -39.00 | ||||||
| -3 | 1 | -3.00 | 171.00 | ||||||
| -5 | 1 | -5.00 | 1205.00 | ||||||
| -9 | 1 | -9.00 | 10161.00 | ||||||
| -15 | 1 | -15.00 | 67455.00 | ||||||
| -45 | 1 | -45.00 | 4556205.00 | ||||||
| 1 | 1 | 1.00 | -49.00 | ||||||
| 3 | 1 | 3.00 | -99.00 | ||||||
| 5 | 1 | 5.00 | -45.00 | ||||||
| 9 | 1 | 9.00 | 2871.00 | ||||||
| 15 | 1 | 15.00 | 33705.00 | ||||||
| 45 | 1 | 45.00 | 3644955.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
5.3 Adding up the two equivalent fractions
(x4-5x3-45) - (30 • x2) x4 - 5x3 - 30x2 - 45
——————————————————————— = ————————————————————
x2 x2
Checking for a perfect cube :
5.4 x4 - 5x3 - 30x2 - 45 is not a perfect cube
Trying to factor by pulling out :
5.5 Factoring: x4 - 5x3 - 30x2 - 45
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -30x2 - 45
Group 2: -5x3 + x4
Pull out from each group separately :
Group 1: (2x2 + 3) • (-15)
Group 2: (x - 5) • (x3)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
5.6 Find roots (zeroes) of : F(x) = x4 - 5x3 - 30x2 - 45
See theory in step 4.2
In this case, the Leading Coefficient is 1 and the Trailing Constant is -45.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,5 ,9 ,15 ,45
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -69.00 | ||||||
| -3 | 1 | -3.00 | -99.00 | ||||||
| -5 | 1 | -5.00 | 455.00 | ||||||
| -9 | 1 | -9.00 | 7731.00 | ||||||
| -15 | 1 | -15.00 | 60705.00 | ||||||
| -45 | 1 | -45.00 | 4495455.00 | ||||||
| 1 | 1 | 1.00 | -79.00 | ||||||
| 3 | 1 | 3.00 | -369.00 | ||||||
| 5 | 1 | 5.00 | -795.00 | ||||||
| 9 | 1 | 9.00 | 441.00 | ||||||
| 15 | 1 | 15.00 | 26955.00 | ||||||
| 45 | 1 | 45.00 | 3584205.00 |
Polynomial Roots Calculator found no rational roots
Final result :
x4 - 5x3 - 30x2 - 45
————————————————————
x2
How did we do?
Please leave us feedback.