Please help with factoring polynomials: Factor 81x^3-3000 and 250 + 2d^3

[confused101]

Could someone please show me the steps, and where you're getting the number from for each step for this:
Factor
81x^3-3000
and or this one
250 + 2d^3

 

[tkhunny]

Find common factors in the individual terms.

81x^2 = 3 * 27 * x^3
3000 = 3 * 1,000

See any common factors?

It seems you are also expected to know how to factor a "Difference of Cubes".

 

[Ebba Sen Pai]

Hi Confused.
With the first expression

81x³-3000 ​

We see that we are dealing with some very large numbers. As Tkhunny said, we must find the GCF of 81 and 3000 by asking what factors they share. As it turns, they share a common factor of 3.
[MATH]81\div 3 = 27[/MATH] and [MATH]3000\div 3 = 1000[/MATH] . This means we can factor out 3 from both terms. Thus we will write

3(27x³-1000)​

Next comes the Difference of Cubes that Tkhunny described. What we must first do is ensure we clearly understand which term is a in our expression and which is b, because subtraction is non commutative and the order here does matter. In other words, if you plug in our "a" which is 27 into the "b" in our formula, we will receive the wrong answer. The formula for the difference of cubes is

​

We know
a=27
b=1000

Plugging in our known values into the formula will produce the answer we need. Sometimes we may need to further factor of the value(s) inside the parenthesis, but it will not be necesary here as it cannot be factored further.

 

[Harry_the_cat]

Ebba Sen Pai wrote:

We know
a=27
b=1000

Plugging in our known values into the formula will produce the answer we need. Sometimes we may need to further factor of the value(s) inside the parenthesis, but it will not be necesary here as it cannot be factored further.

But that's not correct.

Considering \(\displaystyle 27x^3-1000 = (3x)^3 -10^3\) as the difference of two cubes
then
a =3x
and b = 10.

These are the values to substitute into the RHS of the "difference of two cubes" formula.

 

[Ebba Sen Pai]

Ebba Sen Pai wrote:

But that's not correct.

Considering \(\displaystyle 27x^3-1000 = (3x)^3 -10^3\) as the difference of two cubes
then
a =3x
and b = 10.

These are the values to substitute into the RHS of the "difference of two cubes" formula.

Deepest apologies. That was a silly mistake to make. I can't seem to edit my original post, so I will acknowledge the mistake here. I am new to using Latex and was focusing more than I should have on setting things in relatively straightforward visual order and lost the plot towards the end.
Sorry, Henry the Cat and & Confused.

 

[topsquark]

Deepest apologies. That was a silly mistake to make. I can't seem to edit my original post, so I will acknowledge the mistake here. I am new to using Latex and was focusing more than I should have on setting things in relatively straightforward visual order and lost the plot towards the end.
Sorry, Henry the Cat and & Confused.

Don't be too worried about it. And please stop with apologizing! We've all done it and you'll give Harry_the_cat a big head! ?

-Dan