...........help?

Question

find the missing factor

16x^3-54=2(2x-3)(?)

thanks!

Answer 1

16x^3-54=2(2x-3)(y)
y = (16x^3-54)/(2(2x - 3))
y = (8x^3 - 54) / (2x - 3)

You need to divide 8x^3 - 27 by 2x - 3, to get the missing factor.

The first part of the answer is the term with highest degree of x in the numerator (8x^3) divided by the term with highest degree of x in the denominator (2x):

8x^3 / 2x = 4x^2

Multiply that (4x^2) by the denominator (2x-3), and then subtract that from the numerator:

4x^2 * (2x - 3) = 8x^3 - 12x^2

8x^3 - 27 - (8x^3 - 12x^2) =
12x^2 - 27

The second part of the answer is the term with highest degree of x in what is left of the numerator (12x^2) divided by the term with highest degree of x in the denominator (2x):

12x^2 / 2x = 6x

Multiply that (6x) by the denominator (2x-3), and then subtract that from the numerator:

6x * (2x - 3) = 12x^2 - 18x

12x^2 - 27 - (12x^2 - 18x) =
18x - 27

The third part of the answer is the term with highest degree of x in what is left of the numerator (18x) divided by the term with highest degree of x in the denominator (2x):

18x / 2x = 9

Multiply that (9) by the denominator (2x-3), and then subtract that from the numerator:

9 * (2x - 3) = 18x - 27

18x - 27 - (18x - 27) =
0

It divides evenly, since you end up with a remainder of 0. Now, you collect up those three pieces above: 4x^2 + 6x + 9, and that is what you get when you divide (8x^3 - 27) by (2x - 3).

And that's the missing factor you were looking for:
4x^2 + 6x + 9

Answer 2

16x^3-54=2(2x-3)(?)

First divide both sides by 2:
8x^3 - 27 = (2x-3)(?)

This is a difference of cubes.
In general,
(a^3 - b^3) = (a - b)(a^2 + ab + b^2)

In this case:
a^3 = 8x^3 sp a = 2x
b^3 = 27 so b = 3

(a - b)(a^2 + ab + b^2)
=(2x-3)(4x^2 + 6x + 9)

Answer 3

16x^3-54 = 2 (8x^3 - 27)
= 2 [ (2x)^3 - 3^3]
= 2 [ (2x -3) ( 4x^2 + (2x)(3) + 3^2]
= 2 (2x-3) (4x^2 + 6x + 9)

formula:
a^3 - b^3 = (a-b) (a^2 + ab+b^2)

Answer 4

dividing out the first 2 results in

8x^3-27

This factors as

(2x-3)*(4x^2 + 6x +9)

so the missing term is (4x^2 + 6x + 9)

you could infer this result also by comparing the order of the polynomials, so you know the missing term would be of order 2. Also, you could use long division knowing that one root is 3/2

3/2 | 4 0 0 -27/2
...........0 6 9 27/2
-----------------------
...........4 6 9 0 => 4x^2 + 6x + 9

Answer 5

16x^3-54=2(2x-3)(4x^2+6x+9)
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A^3-B^3=(A-B)(A^2+AB+B^2)
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