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 :
h
Simplify ——
h2
Dividing exponential expressions :
1.1 h1 divided by h2 = h(1 - 2) = h(-1) = 1/h1 = 1/h
Equation at the end of step 1 :
4 1 ((((————-4h)-5)+(4•—))-4h)-5 (h2) hStep 2 :
4 Simplify —— h2
Equation at the end of step 2 :
4 4
((((—— - 4h) - 5) + —) - 4h) - 5
h2 h
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using h2 as the denominator :
4h 4h • h2
4h = —— = ———————
1 h2
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 :
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:
4 - (4h • h2) 4 - 4h3
————————————— = ———————
h2 h2
Equation at the end of step 3 :
(4 - 4h3) 4
(((————————— - 5) + —) - 4h) - 5
h2 h
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using h2 as the denominator :
5 5 • h2
5 = — = ——————
1 h2
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
4 - 4h3 = -4 • (h3 - 1)
Trying to factor as a Difference of Cubes:
5.2 Factoring: h3 - 1
Theory : A difference 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 : 1 is the cube of 1
Check : h3 is the cube of h1
Factorization is :
(h - 1) • (h2 + h + 1)
Trying to factor by splitting the middle term
5.3 Factoring h2 + h + 1
The first term is, h2 its coefficient is 1 .
The middle term is, +h its coefficient is 1 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 1 • 1 = 1
Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is 1 .
| -1 | + | -1 | = | -2 | ||
| 1 | + | 1 | = | 2 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
-4 • (h-1) • (h2+h+1) - (5 • h2) -4h3 - 5h2 + 4
———————————————————————————————— = ——————————————
h2 h2
Equation at the end of step 5 :
(-4h3 - 5h2 + 4) 4
((———————————————— + —) - 4h) - 5
h2 h
Step 6 :
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
-4h3 - 5h2 + 4 = -1 • (4h3 + 5h2 - 4)
Polynomial Roots Calculator :
7.2 Find roots (zeroes) of : F(h) = 4h3 + 5h2 - 4
Polynomial Roots Calculator is a set of methods aimed at finding values of h for which F(h)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers h 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 -4.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,2 ,4
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -3.00 | ||||||
| -1 | 2 | -0.50 | -3.25 | ||||||
| -1 | 4 | -0.25 | -3.75 | ||||||
| -2 | 1 | -2.00 | -16.00 | ||||||
| -4 | 1 | -4.00 | -180.00 | ||||||
| 1 | 1 | 1.00 | 5.00 | ||||||
| 1 | 2 | 0.50 | -2.25 | ||||||
| 1 | 4 | 0.25 | -3.62 | ||||||
| 2 | 1 | 2.00 | 48.00 | ||||||
| 4 | 1 | 4.00 | 332.00 |
Polynomial Roots Calculator found no rational roots
Calculating the Least Common Multiple :
7.3 Find the Least Common Multiple
The left denominator is : h2
The right denominator is : h
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| h | 2 | 1 | 2 |
Least Common Multiple:
h2
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 = 1
Right_M = L.C.M / R_Deno = h
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. (-4h3-5h2+4) —————————————————— = ———————————— L.C.M h2 R. Mult. • R. Num. 4 • h —————————————————— = ————— L.C.M h2
Adding fractions that have a common denominator :
7.6 Adding up the two equivalent fractions
(-4h3-5h2+4) + 4 • h -4h3 - 5h2 + 4h + 4
———————————————————— = ———————————————————
h2 h2
Equation at the end of step 7 :
(-4h3 - 5h2 + 4h + 4)
(————————————————————— - 4h) - 5
h2
Step 8 :
Rewriting the whole as an Equivalent Fraction :
8.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using h2 as the denominator :
4h 4h • h2
4h = —— = ———————
1 h2
Checking for a perfect cube :
8.2 -4h3 - 5h2 + 4h + 4 is not a perfect cube
Trying to factor by pulling out :
8.3 Factoring: -4h3 - 5h2 + 4h + 4
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -5h2 + 4
Group 2: -4h3 + 4h
Pull out from each group separately :
Group 1: (-5h2 + 4) • (1) = (5h2 - 4) • (-1)
Group 2: (h2 - 1) • (-4h)
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(h) = -4h3 - 5h2 + 4h + 4
See theory in step 7.2
In this case, the Leading Coefficient is -4 and the Trailing Constant is 4.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
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 | 2 | -0.50 | 1.25 | ||||||
| -1 | 4 | -0.25 | 2.75 | ||||||
| -2 | 1 | -2.00 | 8.00 | ||||||
| -4 | 1 | -4.00 | 164.00 | ||||||
| 1 | 1 | 1.00 | -1.00 | ||||||
| 1 | 2 | 0.50 | 4.25 | ||||||
| 1 | 4 | 0.25 | 4.62 | ||||||
| 2 | 1 | 2.00 | -40.00 | ||||||
| 4 | 1 | 4.00 | -316.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
8.5 Adding up the two equivalent fractions
(-4h3-5h2+4h+4) - (4h • h2) -8h3 - 5h2 + 4h + 4
——————————————————————————— = ———————————————————
h2 h2
Equation at the end of step 8 :
(-8h3 - 5h2 + 4h + 4)
————————————————————— - 5
h2
Step 9 :
Rewriting the whole as an Equivalent Fraction :
9.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using h2 as the denominator :
5 5 • h2
5 = — = ——————
1 h2
Checking for a perfect cube :
9.2 -8h3 - 5h2 + 4h + 4 is not a perfect cube
Trying to factor by pulling out :
9.3 Factoring: -8h3 - 5h2 + 4h + 4
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -5h2 + 4
Group 2: -8h3 + 4h
Pull out from each group separately :
Group 1: (-5h2 + 4) • (1) = (5h2 - 4) • (-1)
Group 2: (2h2 - 1) • (-4h)
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 :
9.4 Find roots (zeroes) of : F(h) = -8h3 - 5h2 + 4h + 4
See theory in step 7.2
In this case, the Leading Coefficient is -8 and the Trailing Constant is 4.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8
of the Trailing Constant : 1 ,2 ,4
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 3.00 | ||||||
| -1 | 2 | -0.50 | 1.75 | ||||||
| -1 | 4 | -0.25 | 2.81 | ||||||
| -1 | 8 | -0.12 | 3.44 | ||||||
| -2 | 1 | -2.00 | 40.00 | ||||||
| -4 | 1 | -4.00 | 420.00 | ||||||
| 1 | 1 | 1.00 | -5.00 | ||||||
| 1 | 2 | 0.50 | 3.75 | ||||||
| 1 | 4 | 0.25 | 4.56 | ||||||
| 1 | 8 | 0.12 | 4.41 | ||||||
| 2 | 1 | 2.00 | -72.00 | ||||||
| 4 | 1 | 4.00 | -572.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
9.5 Adding up the two equivalent fractions
(-8h3-5h2+4h+4) - (5 • h2) -8h3 - 10h2 + 4h + 4
—————————————————————————— = ————————————————————
h2 h2
Step 10 :
Pulling out like terms :
10.1 Pull out like factors :
-8h3 - 10h2 + 4h + 4 =
-2 • (4h3 + 5h2 - 2h - 2)
Checking for a perfect cube :
10.2 4h3 + 5h2 - 2h - 2 is not a perfect cube
Trying to factor by pulling out :
10.3 Factoring: 4h3 + 5h2 - 2h - 2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -2h - 2
Group 2: 4h3 + 5h2
Pull out from each group separately :
Group 1: (h + 1) • (-2)
Group 2: (4h + 5) • (h2)
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(h) = 4h3 + 5h2 - 2h - 2
See theory in step 7.2
In this case, the Leading Coefficient is 4 and the Trailing Constant is -2.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,2
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 1.00 | ||||||
| -1 | 2 | -0.50 | -0.25 | ||||||
| -1 | 4 | -0.25 | -1.25 | ||||||
| -2 | 1 | -2.00 | -10.00 | ||||||
| 1 | 1 | 1.00 | 5.00 | ||||||
| 1 | 2 | 0.50 | -1.25 | ||||||
| 1 | 4 | 0.25 | -2.12 | ||||||
| 2 | 1 | 2.00 | 46.00 |
Polynomial Roots Calculator found no rational roots
Final result :
-2 • (4h3 + 5h2 - 2h - 2)
—————————————————————————
h2
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