Can $15x^3 + 7x^2 - 8x - 27$ be written as a product of two polynomials with integer coefficients? Explain.

Question

i tried factoring it like this :

$x$ $(15x^2 + 7x - 8) - 27$

$x$ $(15x^2 + 15x - 8x - 8) -27$

$x$ $(15x (x+1) - 8(x+1))-27$

$x$ $((x+1)(15x-8)) - 27$

I don't know if it could be factored any further, and i also don't know how to explain.

And i'm sorry i'm learning this on my own so i'm kinda lost, why is the question about integer coefficients? i'm trying to do an exercise about introduction to Direct Proof and Counterexample in a discrete mathematics textbook and this question came right after a question about factoring and parity.

Answer 1

If can we have a factor $x-\frac{m}{n},$ where $gcd(m,n)=1$ and $27$ is divided by $m$ and $15$ is divided by $n$, which is impossible.

Answer 2

The polynomial $15x^3 + 7x^2 - 8x - 27$ is irreducible over $\mathbb{Z}$, since it is irreducible over $\mathbb{F}_2$. So we cannot factor it.