Roots of polynomial equations — common mistakes

Always check for missing terms in the polynomial. For example, in $2x^4 + 3x^2 - 5 = 0$, the coefficients of $x^3$ and $x$ are both zero. So, $a=2, b=0, c=3, d=0, e=-5$. This is a very common place to lose marks.

When performing a substitution for a transformation of roots, be extremely careful with your algebraic expansion, particularly with negative signs. It is often the source of errors. Work systematically and double-check each term.

A missing term simply means its coefficient is zero. In this case, the equation is $1x^3 + 0x^2 + 4x - 2 = 0$, so $a=1, b=0, c=4, d=-2$. This would mean, for example, that the sum of the roots $\Sigma \alpha = -b/a = -0/1 = 0$.

Yes, absolutely. The relationships hold for all roots, whether they are real, rational, irrational or complex. This is a very powerful feature and provides a link to the topic of complex numbers.

The signs simply alternate, starting with a negative. The sum of roots ($\Sigma \alpha$) is $-b/a$. The sum of products in pairs ($\Sigma \alpha\beta$) is $+c/a$. The sum of products in threes ($\Sigma \alpha\beta\gamma$) is $-d/a$, and so on. It's a simple, predictable pattern.

Yes, but it's more complex and less commonly required in exams than $\Sigma \alpha^2$. It can be derived from the identity $(\Sigma \alpha)(\Sigma \alpha^2) = \Sigma \alpha^3 + \dots$. A more direct method is to use the fact that each root satisfies the original equation. For a root $\alpha$, we have $a\alpha^3+b\alpha^2+c\alpha+d=0$. You can write this for $\beta$ and $\gamma$ too, and then sum the three equations: $a\Sigma\alpha^3 + b\Sigma\alpha^2 + c\Sigma\alpha + 3d = 0$. You can then rearrange for $\Sigma\alpha^3$ since you know all the other sums.