Q.A $12$ metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.

A polynomial of degree $n$ in $x$ is $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ with $a_n\ne0$; the corresponding polynomial equation is $P(x)=0$. A number $c$ with $P(c)=0$ is called a root (or zero) — the two words describe exactly the same thing. The leading coefficient is $a_n$, the leading term is $a_nx^n$, and a polynomial with $a_n=1$ is monic. A polynomial's exponents must be non-negative integers, though its coefficients may be any real or complex number — this is exactly why $3x^{-1}+2$, $5x^{1/2}+1$, and trigonometric expressions like $\cos x-\sin x$ are not polynomials, however polynomial-looking they seem.

For the familiar quadratic $ax^2+bx+c=0$ ($a\ne0$), the discriminant $\Delta=b^2-4ac$ governs the roots via $x=\dfrac{-b\pm\sqrt\Delta}{2a}$: with real $a,b,c$, $\Delta>0$ gives real distinct roots, $\Delta=0$ gives equal real roots, and $\Delta<0$ gives no real roots (a non-real conjugate pair — see Complex Conjugate Root Theorem). …