Solution - Adding, subtracting and finding the least common multiple
Other Ways to Solve
Adding, subtracting and finding the least common multipleStep by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "p2" was replaced by "p^2". 1 more similar replacement(s).
Step 1 :
1
Simplify ———
125
Equation at the end of step 1 :
12 6 1
(((8•(p3))+(——•(p2)))+(——•p))+———
5 25 125
Step 2 :
6
Simplify ——
25
Equation at the end of step 2 :
12 6 1 (((8•(p3))+(——•(p2)))+(——•p))+——— 5 25 125Step 3 :
12 Simplify —— 5
Equation at the end of step 3 :
12 6p 1
(((8•(p3))+(——•p2))+——)+———
5 25 125
Step 4 :
Equation at the end of step 4 :
12p2 6p 1 (((8 • (p3)) + ————) + ——) + ——— 5 25 125Step 5 :
Equation at the end of step 5 :
12p2 6p 1
((23p3 + ————) + ——) + ———
5 25 125
Step 6 :
Rewriting the whole as an Equivalent Fraction :
6.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 5 as the denominator :
23p3 23p3 • 5
23p3 = ———— = ————————
1 5
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 :
6.2 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:
23p3 • 5 + 12p2 40p3 + 12p2
——————————————— = ———————————
5 5
Equation at the end of step 6 :
(40p3 + 12p2) 6p 1
(————————————— + ——) + ———
5 25 125
Step 7 :
Step 8 :
Pulling out like terms :
8.1 Pull out like factors :
40p3 + 12p2 = 4p2 • (10p + 3)
Calculating the Least Common Multiple :
8.2 Find the Least Common Multiple
The left denominator is : 5
The right denominator is : 25
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 5 | 1 | 2 | 2 |
| Product of all Prime Factors | 5 | 25 | 25 |
Least Common Multiple:
25
Calculating Multipliers :
8.3 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 = 5
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
8.4 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. 4p2 • (10p+3) • 5 —————————————————— = ————————————————— L.C.M 25 R. Mult. • R. Num. 6p —————————————————— = —— L.C.M 25
Adding fractions that have a common denominator :
8.5 Adding up the two equivalent fractions
4p2 • (10p+3) • 5 + 6p 200p3 + 60p2 + 6p
—————————————————————— = —————————————————
25 25
Equation at the end of step 8 :
(200p3 + 60p2 + 6p) 1
——————————————————— + ———
25 125
Step 9 :
Step 10 :
Pulling out like terms :
10.1 Pull out like factors :
200p3 + 60p2 + 6p = 2p • (100p2 + 30p + 3)
Trying to factor by splitting the middle term
10.2 Factoring 100p2 + 30p + 3
The first term is, 100p2 its coefficient is 100 .
The middle term is, +30p its coefficient is 30 .
The last term, "the constant", is +3
Step-1 : Multiply the coefficient of the first term by the constant 100 • 3 = 300
Step-2 : Find two factors of 300 whose sum equals the coefficient of the middle term, which is 30 .
| -300 | + | -1 | = | -301 | ||
| -150 | + | -2 | = | -152 | ||
| -100 | + | -3 | = | -103 | ||
| -75 | + | -4 | = | -79 | ||
| -60 | + | -5 | = | -65 | ||
| -50 | + | -6 | = | -56 |
For tidiness, printing of 30 lines which failed to find two such factors, was suppressed
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Calculating the Least Common Multiple :
10.3 Find the Least Common Multiple
The left denominator is : 25
The right denominator is : 125
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 5 | 2 | 3 | 3 |
| Product of all Prime Factors | 25 | 125 | 125 |
Least Common Multiple:
125
Calculating Multipliers :
10.4 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 = 5
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
10.5 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. 2p • (100p2+30p+3) • 5 —————————————————— = —————————————————————— L.C.M 125 R. Mult. • R. Num. 1 —————————————————— = ——— L.C.M 125
Adding fractions that have a common denominator :
10.6 Adding up the two equivalent fractions
2p • (100p2+30p+3) • 5 + 1 1000p3 + 300p2 + 30p + 1
—————————————————————————— = ————————————————————————
125 125
Checking for a perfect cube :
10.7 Factoring: 1000p3 + 300p2 + 30p + 1
.
1000p3 + 300p2 + 30p + 1 is a perfect cube which means it is the cube of another polynomial
In our case, the cubic root of 1000p3 + 300p2 + 30p + 1 is 10p + 1
Factorization is (10p + 1)3
Final result :
(10p + 1)3
——————————
125
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