determine productively 54x^4+27x^3-16x
Answers
Answer:
(3x+2)⋅(9x
2
−6x+4)⋅(2x−1)
See steps
Step by Step Solution:
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STEP
1
:
Equation at the end of step 1
(((54 • (x4)) - 33x3) + 16x) - 8
STEP
2
:
Equation at the end of step
2
:
(((2•33x4) - 33x3) + 16x) - 8
STEP
3
:
Checking for a perfect cube
3.1 54x4-27x3+16x-8 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 54x4-27x3+16x-8
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 16x-8
Group 2: 54x4-27x3
Pull out from each group separately :
Group 1: (2x-1) • (8)
Group 2: (2x-1) • (27x3)
-------------------
Add up the two groups :
(2x-1) • (27x3+8)
Which is the desired factorization
Trying to factor as a Sum of Cubes:
3.3 Factoring: 27x3+8
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 : 27 is the cube of 3
Check : 8 is the cube of 2
Check : x3 is the cube of x1
Factorization is :
(3x + 2) • (9x2 - 6x + 4)
Trying to factor by splitting the middle term
3.4 Factoring 9x2 - 6x + 4
The first term is, 9x2 its coefficient is 9 .
The middle term is, -6x its coefficient is -6 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 9 • 4 = 36
Step-2 : Find two factors of 36 whose sum equals the coefficient of the middle term, which is -6 .
-36 + -1 = -37
-18 + -2 = -20
-12 + -3 = -15
-9 + -4 = -13
-6 + -6 = -12
-4 + -9 = -13
For tidiness, printing of 12 lines which failed to find two such factors, was suppressed
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(3x + 2) • (9x2 - 6x + 4) • (2x
Step-by-step explanation:
Factoring: 54x4-27x3+16x-8
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 16x-8
Group 2: 54x⁴-27x³
Pull out from each group separately :
Group 1: (2x-1) • (8)
Group 2: (2x-1) • (27x³)
-------------------
Add up the two groups :
(2x-1) • (27x³+8)