Math, asked by aniscse5bu, 7 months ago

determine productively 54x^4+27x^3-16x

Answers

Answered by shankarpanditkhg
1

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

Answered by biswasshubhmoy
1

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)

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