Solution - Adding, subtracting and finding the least common multiple
Other Ways to Solve
Adding, subtracting and finding the least common multipleStep by Step Solution
Step 1 :
5
Simplify ——
x2
Equation at the end of step 1 :
x 5
((————+4x)-5)+((——+4x)-5)
(x2) x2
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Adding a whole to a fraction
Rewrite the whole as a fraction using x2 as the denominator :
4x 4x • 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
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:
5 + 4x • x2 4x3 + 5
——————————— = ———————
x2 x2
Equation at the end of step 2 :
x (4x3+5)
((————+4x)-5)+(———————-5)
(x2) x2
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x2 as the denominator :
5 5 • x2
5 = — = ——————
1 x2
Trying to factor as a Sum of Cubes :
3.2 Factoring: 4x3 + 5
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 : 4 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
3.3 Find roots (zeroes) of : F(x) = 4x3 + 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 4 and the Trailing Constant is 5.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,5
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 1.00 | ||||||
| -1 | 2 | -0.50 | 4.50 | ||||||
| -1 | 4 | -0.25 | 4.94 | ||||||
| -5 | 1 | -5.00 | -495.00 | ||||||
| -5 | 2 | -2.50 | -57.50 | ||||||
| -5 | 4 | -1.25 | -2.81 | ||||||
| 1 | 1 | 1.00 | 9.00 | ||||||
| 1 | 2 | 0.50 | 5.50 | ||||||
| 1 | 4 | 0.25 | 5.06 | ||||||
| 5 | 1 | 5.00 | 505.00 | ||||||
| 5 | 2 | 2.50 | 67.50 | ||||||
| 5 | 4 | 1.25 | 12.81 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
3.4 Adding up the two equivalent fractions
(4x3+5) - (5 • x2) 4x3 - 5x2 + 5
—————————————————— = —————————————
x2 x2
Equation at the end of step 3 :
x (4x3-5x2+5) ((————+4x)-5)+——————————— (x2) x2Step 4 :
x Simplify —— x2
Dividing exponential expressions :
4.1 x1 divided by x2 = x(1 - 2) = x(-1) = 1/x1 = 1/x
Equation at the end of step 4 :
1 (4x3 - 5x2 + 5)
((— + 4x) - 5) + ———————————————
x x2
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Adding a whole to a fraction
Rewrite the whole as a fraction using x as the denominator :
4x 4x • x
4x = —— = ——————
1 x
Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions
1 + 4x • x 4x2 + 1
—————————— = ———————
x x
Equation at the end of step 5 :
(4x2 + 1) (4x3 - 5x2 + 5)
(————————— - 5) + ———————————————
x x2
Step 6 :
Rewriting the whole as an Equivalent Fraction :
6.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x as the denominator :
5 5 • x
5 = — = —————
1 x
Polynomial Roots Calculator :
6.2 Find roots (zeroes) of : F(x) = 4x2 + 1
See theory in step 3.3
In this case, the Leading Coefficient is 4 and the Trailing Constant is 1.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 5.00 | ||||||
| -1 | 2 | -0.50 | 2.00 | ||||||
| -1 | 4 | -0.25 | 1.25 | ||||||
| 1 | 1 | 1.00 | 5.00 | ||||||
| 1 | 2 | 0.50 | 2.00 | ||||||
| 1 | 4 | 0.25 | 1.25 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
6.3 Adding up the two equivalent fractions
(4x2+1) - (5 • x) 4x2 - 5x + 1
————————————————— = ————————————
x x
Equation at the end of step 6 :
(4x2 - 5x + 1) (4x3 - 5x2 + 5)
—————————————— + ———————————————
x x2
Step 7 :
Trying to factor by splitting the middle term
7.1 Factoring 4x2-5x+1
The first term is, 4x2 its coefficient is 4 .
The middle term is, -5x its coefficient is -5 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 4 • 1 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is -5 .
| -4 | + | -1 | = | -5 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and -1
4x2 - 4x - 1x - 1
Step-4 : Add up the first 2 terms, pulling out like factors :
4x • (x-1)
Add up the last 2 terms, pulling out common factors :
1 • (x-1)
Step-5 : Add up the four terms of step 4 :
(4x-1) • (x-1)
Which is the desired factorization
Polynomial Roots Calculator :
7.2 Find roots (zeroes) of : F(x) = 4x3-5x2+5
See theory in step 3.3
In this case, the Leading Coefficient is 4 and the Trailing Constant is 5.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,5
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -4.00 | ||||||
| -1 | 2 | -0.50 | 3.25 | ||||||
| -1 | 4 | -0.25 | 4.62 | ||||||
| -5 | 1 | -5.00 | -620.00 | ||||||
| -5 | 2 | -2.50 | -88.75 | ||||||
| -5 | 4 | -1.25 | -10.62 | ||||||
| 1 | 1 | 1.00 | 4.00 | ||||||
| 1 | 2 | 0.50 | 4.25 | ||||||
| 1 | 4 | 0.25 | 4.75 | ||||||
| 5 | 1 | 5.00 | 380.00 | ||||||
| 5 | 2 | 2.50 | 36.25 | ||||||
| 5 | 4 | 1.25 | 5.00 |
Polynomial Roots Calculator found no rational roots
Calculating the Least Common Multiple :
7.3 Find the Least Common Multiple
The left denominator is : x
The right denominator is : x2
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| x | 1 | 2 | 2 |
Least Common Multiple:
x2
Calculating Multipliers :
7.4 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = x
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
7.5 Rewrite the two fractions into equivalent fractions
Two fractions are called equivalent if they have the same numeric value.
For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
L. Mult. • L. Num. (x-1) • (4x-1) • x —————————————————— = —————————————————— L.C.M x2 R. Mult. • R. Num. (4x3-5x2+5) —————————————————— = ——————————— L.C.M x2
Adding fractions that have a common denominator :
7.6 Adding up the two equivalent fractions
(x-1) • (4x-1) • x + (4x3-5x2+5) 8x3 - 10x2 + x + 5
———————————————————————————————— = ——————————————————
x2 x2
Checking for a perfect cube :
7.7 8x3 - 10x2 + x + 5 is not a perfect cube
Trying to factor by pulling out :
7.8 Factoring: 8x3 - 10x2 + x + 5
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x + 5
Group 2: 8x3 - 10x2
Pull out from each group separately :
Group 1: (x + 5) • (1)
Group 2: (4x - 5) • (2x2)
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 :
7.9 Find roots (zeroes) of : F(x) = 8x3 - 10x2 + x + 5
See theory in step 3.3
In this case, the Leading Coefficient is 8 and the Trailing Constant is 5.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8
of the Trailing Constant : 1 ,5
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -14.00 | ||||||
| -1 | 2 | -0.50 | 1.00 | ||||||
| -1 | 4 | -0.25 | 4.00 | ||||||
| -1 | 8 | -0.12 | 4.70 | ||||||
| -5 | 1 | -5.00 | -1250.00 | ||||||
| -5 | 2 | -2.50 | -185.00 | ||||||
| -5 | 4 | -1.25 | -27.50 | ||||||
| -5 | 8 | -0.62 | -1.48 | ||||||
| 1 | 1 | 1.00 | 4.00 | ||||||
| 1 | 2 | 0.50 | 4.00 | ||||||
| 1 | 4 | 0.25 | 4.75 | ||||||
| 1 | 8 | 0.12 | 4.98 | ||||||
| 5 | 1 | 5.00 | 760.00 | ||||||
| 5 | 2 | 2.50 | 70.00 | ||||||
| 5 | 4 | 1.25 | 6.25 | ||||||
| 5 | 8 | 0.62 | 3.67 |
Polynomial Roots Calculator found no rational roots
Final result :
8x3 - 10x2 + x + 5
——————————————————
x2
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