Can someone just check if this method and calculations are correct? Task is: Let $P(x) = -5+a+2x+ax^2+7x^3$ and $x=-\frac{1}{2}$ is its zero. Find other zeros and factorise the polynomial.
I solved the equation $P(-\frac{1}{2})=0$ and got that $a= \frac{11}{2}$. Now, $P(x) = -5+\frac{11}{2}+2x+\frac{11}{2}x^2+7x^3 = \frac{1}{2}+2x+\frac{11}{2}x^2+7x^3 $.
Then, I divided P(x) by $(x+\frac{1}{2})$ so $P(x)=(x+\frac{1}{2})(7x^2+2x+1) =(x+\frac{1}{2})(7x^2+2x+1) $
Now, $7x^2+2x+1$ has complex roots: $-1+ i\sqrt{6} $ and $-1- i\sqrt{6} $, so $P(x) =(x+\frac{1}{2})(x- (-1+ i\sqrt{6})) (x- (-1-i\sqrt{6}))$
Thanks in advance.
$\frac12+2x+\frac{11}2x^2+7x^3 =7(x+\frac12)(x-\frac{−1+i\sqrt{6}}{7})(x-\frac{−1-i\sqrt{6}}{7})$