# Determine by inspection the simple roots of the equation

Good day all,

I am having trouble understanding the question to this example:

"Determine by inspection the simple roots of the equation $f(x)=3x^4−x^3−10x^2−2x+4=0$ and hence, by factorisation, find the rest of its roots."

The section deals with polynomials with r given distinct zeros (alpha_k) could be constructed as the product of factors containing those zeros.

Any assistance or supplemental material to better help in understanding this is greatly appreciated.

• Good day to you as well. Are the numbers after the variable $x$ meant to indicate powers? Also, it seems you have written your polynomial function twice. Is the second supposed to be different? – M47145 May 10 '16 at 3:25
• Try plugging in a few small numbers, like 1, -1, 2, -2... If any of these is a root, you will get zero when plugging it into $f$. – ET-phone-homology May 10 '16 at 3:28
Hint: Always try plugging in $x=0$, $x=1$, and $x=-1$ since these are especially easy to evaluate. If you find that $x=k$ is a solution, then you know $(x-k)$ is a factor, so you can divide it out to reduce the degree of the polynomial.
It is easy to see $x=0$ is not a solution since there is a constant in the equation.
Next check for $x=1$ and $x=-1$ by checking sums of all coefficients and sum of alternate coefficients. Turns out, the sum of alternate coefficients is the same (3-10+4 = -1 -2 = -3). Hence $x=-1$ is a solution.
Divide throughout by $(x+1)$ to get the reduced polynomial which gives the other 2 roots.