Factor this polynomial: 8x³ + 36x²y + 54xy² + 27y³?

Question

I can't find the factored form after many tries. can anybody help? thanks.

Answer 1

It is a special form: (a+b)^3=a^3+3*a^2*b+3*a*b^2+b^3. So then 8x³ + 36x²y + 54xy² + 27y³=(2x)^3+3*(2x)^2*3y+2*2x*(3y)^2+(3y)^3=(2x+3y)^3

You get hints in the problem: first off 8x^3 is a perfect cube, 8x^3=(2x)^3. Next 27y^3 is a perfect cube, 27y^3=(3y)^3. So now you should be saying "A ha, this is probably one of those special forms" and then you check the rest of it.

Or use computer software, like Maple, Derive, Mathematica

Final answer:(2x+3y)^3

Answer 2

The way to finding it is to look closely at the numbers, factor them into primes and notice, while remembering that (a+b)^3 = a^3+3a^2b+3ab^2+b^3
8=2^3
36=3*(2^2)*3
54=3*2*3^2
27=3^3
so, if we put u=2x and v=3y the polynomial becomes
u^3+3u^2v+3uv^2+v^3
so, it is the same as (u+v)^3=(2x+3y)^3

Answer 3

8x³ + 36x²y + 54xy² + 27y³=(2x+3y)³ this is binomial expansion

Answer 4

Rearranging the polynomial; x^4 + 8x^2 - 9 = x^4 + 9x^2 -x^2 - 9 => x^2 (x^2 + 9) - a million(x^2 + a million) => (x^2 - a million)(x^2 + 9) => (x - a million)(x + a million)(x^2 + 9) (Ans) you may extra factorise yet factors will be imaginary. => (x - a million)(x + a million)(x^2 - ?(-9)^2] => (x - a million)(x + a million)(x + 3i)(x - 3i) the position i = ?(-a million), an imaginary quantity.

Answer 5

This is the easiest factor problem I've ever seen.
The answer is (2x+3y)^3.
There is no need to explain it.
Hint: Pascal triangle (a+b)^3

Answer 6

(2x+3y)^3
=8x^3+27y^3+3*x*y(2x+3y)