Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
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.
Thank you in advance.
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.
Then try again on the remaining 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.
