Step 1 :
v Simplify — vEquation at the end of step 1 :
8 3 (((((—+3)-9v)-————)+3v)-7)-1 v (v2)Step 2 :
3 Simplify —— v2
Equation at the end of step 2 :
8 3 (((((—+3)-9v)-——)+3v)-7)-1 v v2Step 3 :
8 Simplify — vEquation at the end of step 3 :
8 3 (((((—+3)-9v)-——)+3v)-7)-1 v v2Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1Adding a whole to a fraction
Rewrite the whole as a fraction using v as the denominator :
3 3 • v 3 = — = ————— 1 v 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 :
4.2 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:
8 + 3 • v 3v + 8 ————————— = —————— v v Equation at the end of step 4 :
(3v + 8) 3 ((((———————— - 9v) - ——) + 3v) - 7) - 1 v v2 Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1Subtracting a whole from a fraction
Rewrite the whole as a fraction using v as the denominator :
9v 9v • v 9v = —— = —————— 1 v Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions
(3v+8) - (9v • v) -9v2 + 3v + 8 ————————————————— = ————————————— v v Equation at the end of step 5 :
(-9v2 + 3v + 8) 3 (((——————————————— - ——) + 3v) - 7) - 1 v v2 Step 6 :
Trying to factor by splitting the middle term
6.1Factoring -9v2+3v+8
The first term is, -9v2 its coefficient is -9.
The middle term is, +3v its coefficient is 3.
The last term, "the constant", is +8
Step-1 : Multiply the coefficient of the first term by the constant -9•8=-72
Step-2 : Find two factors of -72 whose sum equals the coefficient of the middle term, which is 3.
| -72 | + | 1 | = | -71 | ||
| -36 | + | 2 | = | -34 | ||
| -24 | + | 3 | = | -21 | ||
| -18 | + | 4 | = | -14 | ||
| -12 | + | 6 | = | -6 | ||
| -9 | + | 8 | = | -1 | ||
| -8 | + | 9 | = | 1 | ||
| -6 | + | 12 | = | 6 | ||
| -4 | + | 18 | = | 14 | ||
| -3 | + | 24 | = | 21 | ||
| -2 | + | 36 | = | 34 | ||
| -1 | + | 72 | = | 71 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Calculating the Least Common Multiple :
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| v | 1 | 2 | 2 |
Least Common Multiple:
v2
Calculating Multipliers :
6.3 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=v
Right_M=L.C.M/R_Deno=1
Making Equivalent Fractions :
6.4 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. (-9v2+3v+8) • v —————————————————— = ——————————————— L.C.M v2 R. Mult. • R. Num. 3 —————————————————— = —— L.C.M v2
Adding fractions that have a common denominator :
6.5 Adding up the two equivalent fractions
(-9v2+3v+8) • v - (3) -9v3 + 3v2 + 8v - 3 ————————————————————— = ——————————————————— v2 v2 Equation at the end of step 6 :
(-9v3 + 3v2 + 8v - 3) ((————————————————————— + 3v) - 7) - 1 v2 Step 7 :
Rewriting the whole as an Equivalent Fraction :
7.1Adding a whole to a fraction
Rewrite the whole as a fraction using v2 as the denominator :
3v 3v • v2 3v = —— = ——————— 1 v2 Checking for a perfect cube :
7.2-9v3 + 3v2 + 8v - 3 is not a perfect cube
Trying to factor by pulling out :
7.3 Factoring: -9v3 + 3v2 + 8v - 3
Thoughtfully split the expression at hand into groups, each group having two terms:
Group 1: 8v - 3
Group 2: -9v3 + 3v2
Pull out from each group separately :
Group 1: (8v - 3) • (1)
Group 2: (3v - 1) • (-3v2)
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 :
7.4 Find roots (zeroes) of : F(v) = -9v3 + 3v2 + 8v - 3
Polynomial Roots Calculator is a set of methods aimed at finding values ofvfor which F(v)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers v which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational numberP/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 -9 and the Trailing Constant is -3. The factor(s) are:
of the Leading Coefficient : 1,3 ,9
of the Trailing Constant : 1 ,3 Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 1.00 | ||||||
| -1 | 3 | -0.33 | -5.00 | ||||||
| -1 | 9 | -0.11 | -3.84 | ||||||
| -3 | 1 | -3.00 | 243.00 | ||||||
| 1 | 1 | 1.00 | -1.00 | ||||||
| 1 | 3 | 0.33 | -0.33 | ||||||
| 1 | 9 | 0.11 | -2.09 | ||||||
| 3 | 1 | 3.00 | -195.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
7.5 Adding up the two equivalent fractions
(-9v3+3v2+8v-3) + 3v • v2 -6v3 + 3v2 + 8v - 3 ————————————————————————— = ——————————————————— v2 v2 Equation at the end of step 7 :
(-6v3 + 3v2 + 8v - 3) (————————————————————— - 7) - 1 v2 Step 8 :
Rewriting the whole as an Equivalent Fraction :
8.1Subtracting a whole from a fraction
Rewrite the whole as a fraction using v2 as the denominator :
7 7 • v2 7 = — = —————— 1 v2 Checking for a perfect cube :
8.2-6v3 + 3v2 + 8v - 3 is not a perfect cube
Trying to factor by pulling out :
8.3 Factoring: -6v3 + 3v2 + 8v - 3
Thoughtfully split the expression at hand into groups, each group having two terms:
Group 1: 8v - 3
Group 2: -6v3 + 3v2
Pull out from each group separately :
Group 1: (8v - 3) • (1)
Group 2: (2v - 1) • (-3v2)
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 :
8.4 Find roots (zeroes) of : F(v) = -6v3 + 3v2 + 8v - 3
See theory in step 7.4
In this case, the Leading Coefficient is -6 and the Trailing Constant is -3. The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,3 Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -2.00 | ||||||
| -1 | 2 | -0.50 | -5.50 | ||||||
| -1 | 3 | -0.33 | -5.11 | ||||||
| -1 | 6 | -0.17 | -4.22 | ||||||
| -3 | 1 | -3.00 | 162.00 | ||||||
| -3 | 2 | -1.50 | 12.00 | ||||||
| 1 | 1 | 1.00 | 2.00 | ||||||
| 1 | 2 | 0.50 | 1.00 | ||||||
| 1 | 3 | 0.33 | -0.22 | ||||||
| 1 | 6 | 0.17 | -1.61 | ||||||
| 3 | 1 | 3.00 | -114.00 | ||||||
| 3 | 2 | 1.50 | -4.50 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
8.5 Adding up the two equivalent fractions
(-6v3+3v2+8v-3) - (7 • v2) -6v3 - 4v2 + 8v - 3 —————————————————————————— = ——————————————————— v2 v2 Equation at the end of step 8 :
(-6v3 - 4v2 + 8v - 3) ————————————————————— - 1 v2 Step 9 :
Rewriting the whole as an Equivalent Fraction :
9.1Subtracting a whole from a fraction
Rewrite the whole as a fraction using v2 as the denominator :
1 1 • v2 1 = — = —————— 1 v2 Step 10 :
Pulling out like terms :
Checking for a perfect cube :
10.26v3 + 4v2 - 8v + 3 is not a perfect cube
Trying to factor by pulling out :
10.3 Factoring: 6v3 + 4v2 - 8v + 3
Thoughtfully split the expression at hand into groups, each group having two terms:
Group 1: -8v + 3
Group 2: 4v2 + 6v3
Pull out from each group separately :
Group 1: (-8v + 3) • (1) = (8v - 3) • (-1)
Group 2: (3v + 2) • (2v2)
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 :
10.4 Find roots (zeroes) of : F(v) = 6v3 + 4v2 - 8v + 3
See theory in step 7.4
In this case, the Leading Coefficient is 6 and the Trailing Constant is 3. The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,3 Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 9.00 | ||||||
| -1 | 2 | -0.50 | 7.25 | ||||||
| -1 | 3 | -0.33 | 5.89 | ||||||
| -1 | 6 | -0.17 | 4.42 | ||||||
| -3 | 1 | -3.00 | -99.00 | ||||||
| -3 | 2 | -1.50 | 3.75 | ||||||
| 1 | 1 | 1.00 | 5.00 | ||||||
| 1 | 2 | 0.50 | 0.75 | ||||||
| 1 | 3 | 0.33 | 1.00 | ||||||
| 1 | 6 | 0.17 | 1.81 | ||||||
| 3 | 1 | 3.00 | 177.00 | ||||||
| 3 | 2 | 1.50 | 20.25 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
10.5 Adding up the two equivalent fractions
(-6v3-4v2+8v-3) - (v2) -6v3 - 5v2 + 8v - 3 —————————————————————— = ——————————————————— v2 v2 Step 11 :
Pulling out like terms :
Checking for a perfect cube :
11.26v3 + 5v2 - 8v + 3 is not a perfect cube
Trying to factor by pulling out :
11.3 Factoring: 6v3 + 5v2 - 8v + 3
Thoughtfully split the expression at hand into groups, each group having two terms:
Group 1: -8v + 3
Group 2: 6v3 + 5v2
Pull out from each group separately :
Group 1: (-8v + 3) • (1) = (8v - 3) • (-1)
Group 2: (6v + 5) • (v2)
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 :
11.4 Find roots (zeroes) of : F(v) = 6v3 + 5v2 - 8v + 3
See theory in step 7.4
In this case, the Leading Coefficient is 6 and the Trailing Constant is 3. The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1 ,3 Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 10.00 | ||||||
| -1 | 2 | -0.50 | 7.50 | ||||||
| -1 | 3 | -0.33 | 6.00 | ||||||
| -1 | 6 | -0.17 | 4.44 | ||||||
| -3 | 1 | -3.00 | -90.00 | ||||||
| -3 | 2 | -1.50 | 6.00 | ||||||
| 1 | 1 | 1.00 | 6.00 | ||||||
| 1 | 2 | 0.50 | 1.00 | ||||||
| 1 | 3 | 0.33 | 1.11 | ||||||
| 1 | 6 | 0.17 | 1.83 | ||||||
| 3 | 1 | 3.00 | 186.00 | ||||||
| 3 | 2 | 1.50 | 22.50 |
Polynomial Roots Calculator found no rational roots
Final result :
-6v3 - 5v2 + 8v - 3 ——————————————————— v2 