Factorize :-
x^3 + 3/2x^3 + 3/4x + 1/8
Answers
Answer:
x3+3/2x2+3/4x+1/8
Final result :
(2x + 1)3
—————————
8
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2". 1 more similar replacement(s).
Step by step solution :
Step 1 :
1
Simplify —
8
Equation at the end of step 1 :
3 3 1
(((x3)+(—•(x2)))+(—•x))+—
2 4 8
Step 2 :
3
Simplify —
4
Equation at the end of step 2 :
3 3 1
(((x3)+(—•(x2)))+(—•x))+—
2 4 8
Step 3 :
3
Simplify —
2
Equation at the end of step 3 :
3 3x 1
(((x3) + (— • x2)) + ——) + —
2 4 8
Step 4 :
Equation at the end of step 4 :
3x2 3x 1
(((x3) + ———) + ——) + —
2 4 8
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 2 as the denominator :
x3 x3 • 2
x3 = —— = ——————
1 2
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 :
5.2 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:
x3 • 2 + 3x2 2x3 + 3x2
———————————— = —————————
2 2
Equation at the end of step 5 :
(2x3 + 3x2) 3x 1
(——————————— + ——) + —
2 4 8
Step 6 :
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
2x3 + 3x2 = x2 • (2x + 3)
Calculating the Least Common Multiple :
7.2 Find the Least Common Multiple
The left denominator is : 2
The right denominator is : 4
Number of times each prime factor
appears in the factorization of:
Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
2 1 2 2
Product of all
Prime Factors 2 4 4
Least Common Multiple:
4
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 = 2
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. x2 • (2x+3) • 2
—————————————————— = ———————————————
L.C.M 4
R. Mult. • R. Num. 3x
—————————————————— = ——
L.C.M 4
Adding fractions that have a common denominator :
7.5 Adding up the two equivalent fractions
x2 • (2x+3) • 2 + 3x 4x3 + 6x2 + 3x
———————————————————— = ——————————————
4 4
Equation at the end of step 7 :
(4x3 + 6x2 + 3x) 1
———————————————— + —
4 8
Step 8 :
Step 9 :
Pulling out like terms :
9.1 Pull out like factors :
4x3 + 6x2 + 3x = x • (4x2 + 6x + 3)
Trying to factor by splitting the middle term
9.2 Factoring 4x2 + 6x + 3
The first term is, 4x2 its coefficient is 4 .
The middle term is, +6x its coefficient is 6 .
The last term, "the constant", is +3
Step-1 : Multiply the coefficient of the first term by the constant 4 • 3 = 12
Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is 6 .
-12 + -1 = -13
-6 + -2 = -8
-4 + -3 = -7
-3 + -4 = -7
-2 + -6 = -8
-1 + -12 = -13
1 + 12 = 13
2 + 6 = 8
3 + 4 = 7
4 + 3 = 7
6 + 2 = 8
12 + 1 = 13
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Calculating the Least Common Multiple :
9.3 Find the Least Common Multiple
The left denominator is : 4
The right denominator is : 8
Number of times each prime factor
appears in the factorization of:
Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
2 2 3 3
Product of all
Prime Factors 4 8 8
Least Common Multiple:
8
Calculating Multipliers :
9.4 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 = 2
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
9.5 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. x • (4x2+6x+3) • 2
—————————————————— = ——————————————————
L.C.M 8
R. Mult. • R. Num. 1
—————————————————— = —
L.C.M 8
Adding fractions that have a common denominator :
9.6 Adding up the two equivalent fractions
x • (4x2+6x+3) • 2 + 1 8x3 + 12x2 + 6x + 1
—————————————————————— = ———————————————————
8 8
Checking for a perfect cube :
9.7 Factoring: 8x3 + 12x2 + 6x + 1
.
8x3 + 12x2 + 6x + 1 is a perfect cube which means it is the cube of another polynomial
In our case, the cubic root of 8x3 + 12x2 + 6x + 1 is 2x + 1
Factorization is (2x + 1)3
Final result :
(2x + 1)3
—————————
8