Solution - Adding, subtracting and finding the least common multiple
Step by Step Solution
Step 1 :
6
Simplify ——
a2
Equation at the end of step 1 :
3 4 6
((((—+2)+—)-a)-——)-4
a 2 a2
Step 2 :
2
Simplify —
1
Equation at the end of step 2 :
3 6
((((—+2)+2)-a)-——)-4
a a2
Step 3 :
3
Simplify —
a
Equation at the end of step 3 :
3 6
((((— + 2) + 2) - a) - ——) - 4
a a2
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Adding a whole to a fraction
Rewrite the whole as a fraction using a as the denominator :
2 2 • a
2 = — = —————
1 a
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:
3 + 2 • a 2a + 3
————————— = ——————
a a
Equation at the end of step 4 :
(2a + 3) 6
(((———————— + 2) - a) - ——) - 4
a a2
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Adding a whole to a fraction
Rewrite the whole as a fraction using a as the denominator :
2 2 • a
2 = — = —————
1 a
Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions
(2a+3) + 2 • a 4a + 3
—————————————— = ——————
a a
Equation at the end of step 5 :
(4a + 3) 6
((———————— - a) - ——) - 4
a a2
Step 6 :
Rewriting the whole as an Equivalent Fraction :
6.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using a as the denominator :
a a • a
a = — = —————
1 a
Adding fractions that have a common denominator :
6.2 Adding up the two equivalent fractions
(4a+3) - (a • a) -a2 + 4a + 3
———————————————— = ————————————
a a
Equation at the end of step 6 :
(-a2 + 4a + 3) 6
(—————————————— - ——) - 4
a a2
Step 7 :
Trying to factor by splitting the middle term
7.1 Factoring -a2+4a+3
The first term is, -a2 its coefficient is -1 .
The middle term is, +4a its coefficient is 4 .
The last term, "the constant", is +3
Step-1 : Multiply the coefficient of the first term by the constant -1 • 3 = -3
Step-2 : Find two factors of -3 whose sum equals the coefficient of the middle term, which is 4 .
| -3 | + | 1 | = | -2 | ||
| -1 | + | 3 | = | 2 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Calculating the Least Common Multiple :
7.2 Find the Least Common Multiple
The left denominator is : a
The right denominator is : a2
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| a | 1 | 2 | 2 |
Least Common Multiple:
a2
Calculating Multipliers :
7.3 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 = a
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
7.4 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. (-a2+4a+3) • a —————————————————— = —————————————— L.C.M a2 R. Mult. • R. Num. 6 —————————————————— = —— L.C.M a2
Adding fractions that have a common denominator :
7.5 Adding up the two equivalent fractions
(-a2+4a+3) • a - (6) -a3 + 4a2 + 3a - 6
———————————————————— = ——————————————————
a2 a2
Equation at the end of step 7 :
(-a3 + 4a2 + 3a - 6)
———————————————————— - 4
a2
Step 8 :
Rewriting the whole as an Equivalent Fraction :
8.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using a2 as the denominator :
4 4 • a2
4 = — = ——————
1 a2
Checking for a perfect cube :
8.2 -a3 + 4a2 + 3a - 6 is not a perfect cube
Trying to factor by pulling out :
8.3 Factoring: -a3 + 4a2 + 3a - 6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 3a - 6
Group 2: -a3 + 4a2
Pull out from each group separately :
Group 1: (a - 2) • (3)
Group 2: (a - 4) • (-a2)
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(a) = -a3 + 4a2 + 3a - 6
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 -1 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -4.00 | ||||||
| -2 | 1 | -2.00 | 12.00 | ||||||
| -3 | 1 | -3.00 | 48.00 | ||||||
| -6 | 1 | -6.00 | 336.00 | ||||||
| 1 | 1 | 1.00 | 0.00 | a - 1 | |||||
| 2 | 1 | 2.00 | 8.00 | ||||||
| 3 | 1 | 3.00 | 12.00 | ||||||
| 6 | 1 | 6.00 | -60.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
-a3 + 4a2 + 3a - 6
can be divided with a - 1
Polynomial Long Division :
8.5 Polynomial Long Division
Dividing : -a3 + 4a2 + 3a - 6
("Dividend")
By : a - 1 ("Divisor")
| dividend | - | a3 | + | 4a2 | + | 3a | - | 6 | |
| - divisor | * -a2 | - | a3 | + | a2 | ||||
| remainder | 3a2 | + | 3a | - | 6 | ||||
| - divisor | * 3a1 | 3a2 | - | 3a | |||||
| remainder | 6a | - | 6 | ||||||
| - divisor | * 6a0 | 6a | - | 6 | |||||
| remainder | 0 |
Quotient : -a2+3a+6 Remainder: 0
Trying to factor by splitting the middle term
8.6 Factoring -a2+3a+6
The first term is, -a2 its coefficient is -1 .
The middle term is, +3a its coefficient is 3 .
The last term, "the constant", is +6
Step-1 : Multiply the coefficient of the first term by the constant -1 • 6 = -6
Step-2 : Find two factors of -6 whose sum equals the coefficient of the middle term, which is 3 .
| -6 | + | 1 | = | -5 | ||
| -3 | + | 2 | = | -1 | ||
| -2 | + | 3 | = | 1 | ||
| -1 | + | 6 | = | 5 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Adding fractions that have a common denominator :
8.7 Adding up the two equivalent fractions
(-a2+3a+6) • (a-1) - (4 • a2) -a3 + 3a - 6
————————————————————————————— = ————————————
a2 a2
Step 9 :
Pulling out like terms :
9.1 Pull out like factors :
-a3 + 3a - 6 = -1 • (a3 - 3a + 6)
Polynomial Roots Calculator :
9.2 Find roots (zeroes) of : F(a) = a3 - 3a + 6
See theory in step 8.4
In this case, the Leading Coefficient is 1 and the Trailing Constant is 6.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 8.00 | ||||||
| -2 | 1 | -2.00 | 4.00 | ||||||
| -3 | 1 | -3.00 | -12.00 | ||||||
| -6 | 1 | -6.00 | -192.00 | ||||||
| 1 | 1 | 1.00 | 4.00 | ||||||
| 2 | 1 | 2.00 | 8.00 | ||||||
| 3 | 1 | 3.00 | 24.00 | ||||||
| 6 | 1 | 6.00 | 204.00 |
Polynomial Roots Calculator found no rational roots
Final result :
-a3 + 3a - 6
————————————
a2
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