Help: Factor the following Cubic Polynomial

Hi, I'm having some serious trouble with the below problem.

Factor the following cubic polynomial. Identify parts:

x^3+6x^2y+12xy^2+8y^3=

As you can see there is no GCF for the whole equation, and grouping doesn't work. I am truly lost here and it's for a test on Monday. I know from a previous problem that the final factored form is (x+2y)^3 but I can't seem to get there.

Thanks!

MacaroniMarconi said:

Hi, I'm having some serious trouble with the below problem.

Factor the following cubic polynomial. Identify parts:

x^3+6x^2y+12xy^2+8y^3=

As you can see there is no GCF for the whole equation, and grouping doesn't work. I am truly lost here and it's for a test on Monday. I know from a previous problem that the final factored form is (x+2y)^3 but I can't seem to get there.

Thanks!

Click to expand...

It is a standard formula. You should remember it !!

(a + b)³ = a³ + 3a²b + 3ab² + b³

In your case

a = x and b = 2y

Subhotosh Khan said:

It is a standard formula. You should remember it !!

(a + b)³ = a³ + 3a²b + 3ab² + b³

In your case

a = x and b = 2y

Click to expand...

This is great, thank you so much. I have the formula written down and highlighted now. If you don't mind a quick clarification for me; there are no other steps it's just the (a + b)³ = a³ + 3a²b + 3ab² + b³
and the formula would be the identification that they're asking for?

MacaroniMarconi said:

there are no other steps[?]

it's just the (a + b)³ = a³ + 3a²b + 3ab² + b3

Click to expand...

The other steps come first (i.e., recognizing the pattern of a binomial expansion, and then figuring out the coefficients).

Look at the form of the given expression, written like so and ignoring the coefficients:

x³y⁰ + x²y¹ + x¹y² + x⁰y³

See how the powers of x decrease from 3 to 0, while the powers of y increase from 0 to 3?

This is a clue indicating that the factored form is (x +Ay)^3 for some constant A.

the formula would be the identification that they're asking for?

Click to expand...

I don't know what their meaning is for "Identify parts".

MacaroniMarconi said:

Hi, I'm having some serious trouble with the below problem.

Factor the following cubic polynomial. Identify parts:

x^3+6x^2y+12xy^2+8y^3=

As you can see there is no GCF for the whole equation, and grouping doesn't work. I am truly lost here and it's for a test on Monday. I know from a previous problem that the final factored form is (x+2y)^3 but I can't seem to get there.

Thanks!

Click to expand...

May be the solution should look like:

x³ + 6x²y + 12xy² + 8y³

= x³ + 3 * x² * (2y) + 3 * x * (2y)² + (2y)³

= (x + 2y)³

If I were doing this problem, the first thing I would do is notice that \(\displaystyle 8y^3= (2y)^3\). That would lead me to think in terms of "2y" and then it would be easy to see that \(\displaystyle 12xy^2= 3x(4y^2)= 3x(2y)^2\). And then I would see that \(\displaystyle 6x^2y= 3x^2(2y)\) leading to the "exact cubic" the others have referred to.