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 :
Equation at the end of step 1 :
(15•(a2)) (13a+6)
((10a-—————————)+((12a+9)•(6•(a2))))+(———————•-4)
(4•(a2)) 32a2
Step 2 :
13a + 6
Simplify ———————
32a2
Equation at the end of step 2 :
(15•(a2)) (13a+6)
((10a-—————————)+((12a+9)•(6•(a2))))+(———————•-4)
(4•(a2)) 32a2
Step 3 :
Equation at the end of step 3 :
(15•(a2)) -4•(13a+6) ((10a-—————————)+((12a+9)•(6•(a2))))+—————————— (4•(a2)) 9a2Step 4 :
Equation at the end of step 4 :
(15•(a2)) -4•(13a+6)
((10a-—————————)+((12a+9)•(2•3a2)))+——————————
(4•(a2)) 9a2
Step 5 :
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
12a + 9 = 3 • (4a + 3)
Multiplying exponents :
6.2 31 multiplied by 31 = 3(1 + 1) = 32
Equation at the end of step 6 :
(15•(a2)) -4•(13a+6) ((10a-—————————)+(32•2a2)•(4a+3))+—————————— (4•(a2)) 9a2Step 7 :
Equation at the end of step 7 :
(15•(a2)) -4•(13a+6) ((10a-—————————)+(32•2a2)•(4a+3))+—————————— 22a2 9a2Step 8 :
Equation at the end of step 8 :
(3•5a2) -4•(13a+6)
((10a-———————)+(32•2a2)•(4a+3))+——————————
22a2 9a2
Step 9 :
(3•5a2)
Simplify ———————
22a2
Canceling Out :
9.1 Canceling out a2 as it appears on both sides of the fraction line
Equation at the end of step 9 :
15 -4•(13a+6)
((10a-——)+(32•2a2)•(4a+3))+——————————
4 9a2
Step 10 :
Rewriting the whole as an Equivalent Fraction :
10.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 4 as the denominator :
10a 10a • 4
10a = ——— = ———————
1 4
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 :
10.2 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:
10a • 4 - (15) 40a - 15
—————————————— = ————————
4 4
Equation at the end of step 10 :
(40a - 15) -4 • (13a + 6)
(—————————— + (32•2a2) • (4a + 3)) + ——————————————
4 9a2
Step 11 :
Rewriting the whole as an Equivalent Fraction :
11.1 Adding a whole to a fraction
Rewrite the whole as a fraction using 4 as the denominator :
(32•2a2) • (4a + 3) (32•2a2) • (4a + 3) • 4
(32•2a2) • (4a + 3) = ——————————————————— = ———————————————————————
1 4
Step 12 :
Pulling out like terms :
12.1 Pull out like factors :
40a - 15 = 5 • (8a - 3)
Adding fractions that have a common denominator :
12.2 Adding up the two equivalent fractions
5 • (8a-3) + (32•2a2) • (4a+3) • 4 288a3 + 216a2 + 40a - 15
—————————————————————————————————— = ————————————————————————
4 4
Equation at the end of step 12 :
(288a3 + 216a2 + 40a - 15) -4 • (13a + 6)
—————————————————————————— + ——————————————
4 9a2
Step 13 :
Checking for a perfect cube :
13.1 288a3+216a2+40a-15 is not a perfect cube
Trying to factor by pulling out :
13.2 Factoring: 288a3+216a2+40a-15
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 40a-15
Group 2: 288a3+216a2
Pull out from each group separately :
Group 1: (8a-3) • (5)
Group 2: (4a+3) • (72a2)
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 :
13.3 Find roots (zeroes) of : F(a) = 288a3+216a2+40a-15
Polynomial Roots Calculator is a set of methods aimed at finding values of a for which F(a)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers a 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 288 and the Trailing Constant is -15.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,4 ,6 ,8 ,9 ,12 ,16 ,18 , etc
of the Trailing Constant : 1 ,3 ,5 ,15
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -127.00 | ||||||
| -1 | 2 | -0.50 | -17.00 | ||||||
| -1 | 3 | -0.33 | -15.00 | ||||||
| -1 | 4 | -0.25 | -16.00 | ||||||
| -1 | 6 | -0.17 | -17.00 |
Note - For tidiness, printing of 55 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Calculating the Least Common Multiple :
13.4 Find the Least Common Multiple
The left denominator is : 4
The right denominator is : 9a2
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 2 | 0 | 2 |
| 3 | 0 | 2 | 2 |
| Product of all Prime Factors | 4 | 9 | 36 |
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| a | 0 | 2 | 2 |
Least Common Multiple:
36a2
Calculating Multipliers :
13.5 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 = 9a2
Right_M = L.C.M / R_Deno = 4
Making Equivalent Fractions :
13.6 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. (288a3+216a2+40a-15) • 9a2 —————————————————— = —————————————————————————— L.C.M 36a2 R. Mult. • R. Num. -4 • (13a+6) • 4 —————————————————— = ———————————————— L.C.M 36a2
Adding fractions that have a common denominator :
13.7 Adding up the two equivalent fractions
(288a3+216a2+40a-15) • 9a2 + -4 • (13a+6) • 4 2592a5 + 1944a4 + 360a3 - 135a2 - 208a - 96
————————————————————————————————————————————— = ———————————————————————————————————————————
36a2 36a2
Trying to factor by pulling out :
13.8 Factoring: 2592a5 + 1944a4 + 360a3 - 135a2 - 208a - 96
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 360a3 - 135a2
Group 2: 2592a5 + 1944a4
Group 3: -208a - 96
Pull out from each group separately :
Group 1: (8a - 3) • (45a2)
Group 2: (4a + 3) • (648a4)
Group 3: (13a + 6) • (-16)
Looking for common sub-expressions :
Group 1: (8a - 3) • (45a2)
Group 3: (13a + 6) • (-16)
Group 2: (4a + 3) • (648a4)
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 :
13.9 Find roots (zeroes) of : F(a) = 2592a5 + 1944a4 + 360a3 - 135a2 - 208a - 96
See theory in step 13.3
In this case, the Leading Coefficient is 2592 and the Trailing Constant is -96.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,4 ,6 ,8 ,9 ,12 ,16 ,18 , etc
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,16 ,24 ,32 , etc
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -1031.00 | ||||||
| -1 | 2 | -0.50 | -30.25 | ||||||
| -1 | 3 | -0.33 | -41.67 | ||||||
| -1 | 4 | -0.25 | -53.00 | ||||||
| -1 | 6 | -0.17 | -65.58 |
Note - For tidiness, printing of 61 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Final result :
2592a5 + 1944a4 + 360a3 - 135a2 - 208a - 96
———————————————————————————————————————————
36a2
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