Question

Simplify:
a^2(2b-a^2)-2b(a^3-a^2)+3a^2(a^3-b)
​

Answer 1

Answer:

STEP

1

:

1

Simplify —

2

Equation at the end of step

1

:

(b2) b 1

((((((((a2)+————)-(b3))-a)+(—•(a2)))-2ab)+(2•(b2)))-(—•a))-2b

(a3) 2 2

STEP

2

:

Equation at the end of step

2

:

(b2) b a

((((((((a2)+————)-(b3))-a)+(—•(a2)))-2ab)+2b2)-—)-2b

(a3) 2 2

STEP

3

:

b

Simplify —

2

Equation at the end of step

3

:

(b2) b a

((((((((a2)+————)-(b3))-a)+(—•a2))-2ab)+2b2)-—)-2b

(a3) 2 2

STEP

4

:

Equation at the end of step 4

(b2) a2b a

((((((((a2)+————)-(b3))-a)+———)-2ab)+2b2)-—)-2b

(a3) 2 2

STEP

5

:

b2

Simplify ——

a3

Equation at the end of step

5

:

b2 a2b a

((((((((a2)+——)-b3)-a)+———)-2ab)+2b2)-—)-2b

a3 2 2

STEP

6

:

Rewriting the whole as an Equivalent Fraction :

6.1 Adding a fraction to a whole

Rewrite the whole as a fraction using a3 as the denominator :

a2 a2 • a3

a2 = —— = ———————

1 a3

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:

a2 • a3 + b2 a5 + b2

———————————— = ———————

a3 a3

Equation at the end of step

6

:

(a5+b2) a2b a

((((((———————-b3)-a)+———)-2ab)+2b2)-—)-2b

a3 2 2

STEP

7

:

Rewriting the whole as an Equivalent Fraction

7.1 Subtracting a whole from a fraction

Rewrite the whole as a fraction using a3 as the denominator :

b3 b3 • a3

b3 = —— = ———————

1 a3

Adding fractions that have a common denominator :

7.2 Adding up the two equivalent fractions

(a5+b2) - (b3 • a3) a5 - a3b3 + b2

——————————————————— = ——————————————

a3 a3

Equation at the end of step

7

:

(a5-a3b3+b2) a2b a

(((((————————————-a)+———)-2ab)+2b2)-—)-2b

a3 2 2

STEP

8

:

Rewriting the whole as an Equivalent Fraction

8.1 Subtracting a whole from a fraction

Rewrite the whole as a fraction using a3 as the denominator :

a a • a3

a = — = ——————

1 a3

Trying to factor a multi variable polynomial :

8.2 Factoring a5 - a3b3 + b2

Try to factor this multi-variable trinomial using trial and error

Factorization fails

Adding fractions that have a common denominator :

8.3 Adding up the two equivalent fractions

(a5-a3b3+b2) - (a • a3) a5 - a4 - a3b3 + b2

——————————————————————— = ———————————————————

a3 a3

Equation at the end of step

8

:

(a5-a4-a3b3+b2) a2b a

((((———————————————+———)-2ab)+2b2)-—)-2b

a3 2 2

STEP

9

:

Checking for a perfect cube

9.1 a5-a4-a3b3+b2 is not a perfect cube

Calculating the Least Common Multiple :

9.2 Find the Least Common Multiple

The left denominator is : a3

The right denominator is : 2

Number of times each prime factor

appears in the factorization of:

Prime

Factor Left

Denominator Right

Denominator L.C.M = Max

{Left,Right}

2 0 1 1

Product of all

Prime Factors 1 2 2

Number of times each Algebraic Factor

appears in the factorization of:

Algebraic

Factor Left

Denominator Right

Denominator L.C.M = Max

{Left,Right}

a 3 0 3

Least Common Multiple:

2a3

Calculating Multipliers :

9.3 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 = 2

Right_M = L.C.M / R_Deno = a3

Making Equivalent Fractions

Answer 2

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