Simplify:
(a) 3a^2(a^4 - b^4) - 2ab(a^5 - b^5) + ab^3 (a^3 - b^3)
Answers
Step by Step Solution:
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STEP
1
:
b4
Simplify ——
a2
Equation at the end of step
1
:
b4
(((((a4)+((a3)•b))-(a•(b3)))-——)-2ab)+b2
a2
STEP
2
:
Rewriting the whole as an Equivalent Fraction
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using a2 as the denominator :
a4 + a3b - ab3 (a4 + a3b - ab3) • a2
a4 + a3b - ab3 = —————————————— = —————————————————————
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
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
a4 + a3b - ab3 = a • (a3 + a2b - b3)
Trying to factor a multi variable polynomial :
3.2 Factoring a3 + a2b - b3
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Adding fractions that have a common denominator :
3.3 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:
a • (a3+a2b-b3) • a2 - (b4) a6 + a5b - a3b3 - b4
——————————————————————————— = ————————————————————
a2 a2
Equation at the end of step
3
:
(a6 + a5b - a3b3 - b4)
(—————————————————————— - 2ab) + b2
a2
STEP
4
:
Rewriting the whole as an Equivalent Fraction
4.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using a2 as the denominator :
2ab 2ab • a2
2ab = ——— = ————————
1 a2
Checking for a perfect cube :
4.2 a6 + a5b - a3b3 - b4 is not a perfect cube
Adding fractions that have a common denominator :
4.3 Adding up the two equivalent fractions
(a6+a5b-a3b3-b4) - (2ab • a2) a6 + a5b - a3b3 - 2a3b - b4
————————————————————————————— = ———————————————————————————
a2 a2
Equation at the end of step
4
:
(a6 + a5b - a3b3 - 2a3b - b4)
————————————————————————————— + b2
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 a2 as the denominator :
b2 b2 • a2
b2 = —— = ———————
1 a2
Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions
(a6+a5b-a3b3-2a3b-b4) + b2 • a2 a6 + a5b - a3b3 - 2a3b + a2b2 - b4
——————————————————————————————— = ——————————————————————————————————
a2 a2
Trying to factor by pulling out :
5.3 Factoring: a6 + a5b - a3b3 - 2a3b + a2b2 - b4
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -a3b3 - 2a3b
Group 2: a5b + a6
Group 3: a2b2 - b4
Pull out from each group separately :
Group 1: (b2 + 2) • (-a3b)
Group 2: (a + b) • (a5)
Group 3: (a2 - b2) • (b2)
Looking for common sub-expressions :
Group 1: (b2 + 2) • (-a3b)
Group 3: (a2 - b2) • (b2)
Group 2: (a + b) • (a5)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Final result :
a6 + a5b - a3b3 - 2a3b + a2b2 - b4
——————————————————————————————————
a2
Answer:
1a¹⁶ b¹⁶
=(3a⁶-b⁴) - (2a⁶b⁵) + ( a⁴b³- b³)
= 3a⁶b⁴- 2a⁶b⁶
= 1a¹² 6¹⁰ + (a⁴b⁶)
= 1a¹⁶ b¹⁶
