Math Help - Polynomial Reduction Help

1. Polynomial Reduction Help

Reduce to lowest terms:
$\frac {x^3+x^2+x+1}{x^3+3x^3+3x+1}$

I don't see how this can be factored to be reduced. Thanks.
The answer is $\frac{x^2+1}{x^2+2x+1}$

2. Re: Polynomial Reduction Help

Just to be sure, is that really $3x^3$ for the second term in the denominator?

3. Re: Polynomial Reduction Help

For the numerator:

$f(x)=x^3+x^2+x+1$

You could use the rational roots theorem, which states that if the given polynomial has a rational root, it will come from the list $x=\pm1$. We can see that $x=+1$ cannot work, hence we find:

$f(-1)=-1+1-1+1=0$

So, we know $x+1$ is a factor of $f(x)$. Use of division finds:

$f(x)=(x+1)(x^2+1)$

Now, for the denominator, we should recognize the binomial coefficients arising from the cube of a binomial, i.e.:

$x^3+3x^2+3x+1=(x+1)^3$ and so we may state:

$\frac{x^3+x^2+x+1}{x^3+3x^2+3x+1}=\frac{(x+1)(x^2+ 1)}{(x+1)^3}=\frac{x^2+1}{(x+1)^2}=\frac{x^2+1}{x^ 2+2x+1}$

4. Re: Polynomial Reduction Help

grillage - you are correct it is $3x^2$. Thanks for the help markFL2, I didn't know about the rational roots theorem, that will make working with polynomials a whole lot easier.