solve - 2a ^ 3 * b ^ 2 + a ^ 5 + a * b ^ 4
Answers
Answer:
Final result :
-2a3 - 5ab3 - 3b2
—————————————————
a2
Step-by-step explanation:
STEP
1
:
b
Simplify —
a
Equation at the end of step
1
:
b b
((2a-((3•————)•b))-4a)-((5•—)•b2)
(a2) a
STEP
2
:
Multiplying exponential expressions :
2.1 b1 multiplied by b2 = b(1 + 2) = b3
Equation at the end of step
2
:
b 5b3
((2a-((3•————)•b))-4a)-———
(a2) a
STEP
3
:
b
Simplify ——
a2
Equation at the end of step
3
:
b 5b3
((2a - ((3 • ——) • b)) - 4a) - ———
a2 a
STEP
4
:
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using a2 as the denominator :
2a 2a • a2
2a = —— = ———————
1 a2
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:
2a • a2 - (3b2) 2a3 - 3b2
——————————————— = —————————
a2 a2
Equation at the end of step
4
:
(2a3 - 3b2) 5b3
(——————————— - 4a) - ———
a2 a
STEP
5
:
Rewriting the whole as an Equivalent Fraction :
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using a2 as the denominator :
4a 4a • a2
4a = —— = ———————
1 a2
Trying to factor as a Difference of Cubes:
5.2 Factoring: 2a3 - 3b2
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 : 2 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Adding fractions that have a common denominator :
5.3 Adding up the two equivalent fractions
(2a3-3b2) - (4a • a2) -2a3 - 3b2
————————————————————— = ——————————
a2 a2
Equation at the end of step
5
:
(-2a3 - 3b2) 5b3
———————————— - ———
a2 a
STEP
6
:
STEP
7
:
Pulling out like terms :
7.1 Pull out like factors :
-2a3 - 3b2 = -1 • (2a3 + 3b2)
Trying to factor as a Sum of Cubes:
7.2 Factoring: 2a3 + 3b2
Theory : A sum 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 : 2 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Calculating the Least Common Multiple :
7.3 Find the Least Common Multiple
The left denominator is : a2
The right denominator is : a
Number of times each Algebraic Factor
appears in the factorization of:
Algebraic
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
a 2 1 2
Least Common Multiple:
a2
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 = a
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. (-2a3-3b2)
—————————————————— = ——————————
L.C.M a2
R. Mult. • R. Num. 5b3 • a
—————————————————— = ———————
L.C.M a2
Adding fractions that have a common denominator :
7.6 Adding up the two equivalent fractions
(-2a3-3b2) - (5b3 • a) -2a3 - 5ab3 - 3b2
—————————————————————— = —————————————————
a2 a2
STEP
8
:
Pulling out like terms :
8.1 Pull out like factors :
-2a3 - 5ab3 - 3b2 = -1 • (2a3 + 5ab3 + 3b2)
Trying to factor a multi variable polynomial :
8.2 Factoring 2a3 + 5ab3 + 3b2
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Final result :
-2a3 - 5ab3 - 3b2
—————————————————
a2