Simplify (2a2 + 5b2
)(a –b) + (a2 – b
2
)(3a + 4b)
Answers
Answer:
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "b2" was replaced by "b^2". 3 more similar replacement(s).
STEP
1
:
Equation at the end of step 1
b (b2)
(2a-(((5•———————)•2)•(a2)))-(2•———————————————————————)
(2a+2b) (((6•(a2))+ab)-(3•5b2))
STEP
2
:
Equation at the end of step
2
:
b (b2)
(2a-(((5•———————)•2)•(a2)))-(2•——————————————————————)
(2a+2b) (((2•3a2)+ab)-(3•5b2))
STEP
3
:
b2
Simplify ———————————————
6a2 + ab - 15b2
Trying to factor a multi variable polynomial :
3.1 Factoring 6a2 + ab - 15b2
Try to factor this multi-variable trinomial using trial and error
Found a factorization : (2a - 3b)•(3a + 5b)
Equation at the end of step
3
:
b b2
(2a-(((5•———————)•2)•(a2)))-(2•———————————————)
(2a+2b) (2a-3b)•(3a+5b)
STEP
4
:
Equation at the end of step 4
b 2b2
(2a-(((5•———————)•2)•(a2)))-———————————————
(2a+2b) (2a-3b)•(3a+5b)
STEP
5
:
b
Simplify ———————
2a + 2b
STEP
6
:
Pulling out like terms
6.1 Pull out like factors :
2a + 2b = 2 • (a + b)
Equation at the end of step
6
:
b 2b2
(2a-(((5•———————)•2)•a2))-———————————————
2•(a+b) (2a-3b)•(3a+5b)
STEP
7
:
Equation at the end of step 7
5b 2b2
(2a-((———————•2)•a2))-———————————————
2•(a+b) (2a-3b)•(3a+5b)
STEP
8
:
Equation at the end of step 8
5b 2b2
(2a - (————— • a2)) - —————————————————————
a + b (2a - 3b) • (3a + 5b)
STEP
9
:
Equation at the end of step 9
5a2b 2b2
(2a - —————) - —————————————————————
a + b (2a - 3b) • (3a + 5b)
STEP
10
:
Rewriting the whole as an Equivalent Fraction
10.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using (a+b) as the denominator :
2a 2a • (a + b)
2a = —— = ————————————
1 (a + b)
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:
2a • (a+b) - (5a2b) -5a2b + 2a2 + 2ab
——————————————————— = —————————————————
1 • (a+b) 1 • (a + b)
Equation at the end of step
10
:
(-5a2b + 2a2 + 2ab) 2b2
——————————————————— - —————————————————————
1 • (a + b) (2a - 3b) • (3a + 5b)
STEP
11
:
STEP
12
:
Pulling out like terms :
12.1 Pull out like factors :
-5a2b + 2a2 + 2ab = -a • (5ab - 2a - 2b)
Calculating the Least Common Multiple :
12.2 Find the Least Common Multiple
The left denominator is : a + b
The right denominator is : (2a - 3b) • (3a + 5b)
Number of times each Algebraic Factor
appears in the factorization of:
Algebraic
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
a + b 1 0 1
2a - 3b 0 1 1
3a + 5b 0 1 1
Least Common Multiple:
(a + b) • (2a - 3b) • (3a + 5b)
Calculating Multipliers :
12.3 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 = (2a - 3b)•(3a + 5b)
Right_M = L.C.M / R_Deno = a + b
Making Equivalent Fractions :
12.4 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. -a • (5ab-2a-2b) • (2a-3b) • (3a+5b)
—————————————————— = ————————————————————————————————————
L.C.M (a+b) • (2a-3b) • (3a+5b)
R. Mult. • R. Num. 2b2 • (a+b)
—————————————————— = —————————————————————————
L.C.M (a+b) • (2a-3b) • (3a+5b)
Adding fractions that have a common denominator :
12.5 Adding up the two equivalent fractions
-a • (5ab-2a-2b) • (2a-3b) • (3a+5b) - (2b2 • (a+b)) -30a4b+12a4-5a3b2+14a3b+75a2b3-28a2b2-30ab3-2ab2-2b3
———————————————————————————————————————————————————— = ————————————————————————————————————————————————————
(a+b) • (2a-3b) • (3a+5b) (a+b) • (2a-3b) • (3a+5b)
STEP
13
:
Pulling out like terms :
13.1 Pull out like factors :
-30a4b + 12a4 - 5a3b2 + 14a3b + 75a2b3 - 28a2b2 - 30ab3 - 2ab2 - 2b3 =
-1 • (30a4b - 12a4 + 5a3b2 - 14a3b - 75a2b3 + 28a2b2 + 30ab3 + 2ab2 + 2b3)