Construct a cubic equation with roots $1,1$ and $-2$.

(i) A quadratic equation cannot have more than two roots.

(ii) If $a{x}^2+bx+c=0,a\ne 0$ is a quadratic equation with real coefficients, then its roots $\alpha$ and $\beta$ given by $\alpha =\frac{-b+\sqrt{D}}{2a}$ and $\beta =\frac{-b-\sqrt{D}}{2a}$, where $D=b^2-4ac$ is the discriminant of the quadratic equation.

(iii) A quadratic equation whose roots are $\alpha$ and $\beta$ is $x^2-(\alpha +\beta )x+\alpha \beta =0$; that is, a quadratic equation with given roots is, $x^2-\text{(sum of the roots)} x+\text{(product of the roots)}=0.$

2. Nature of roots of a quadratic equation:

(i) If $D=0,$ then roots are real and equal.

(ii) If $a,b,c\in Q$ and $D$ is positive and a perfect square, then roots are rational and unequal.

(iii) If $a,b,c\in R$ and $D$ is positive and a perfect square, then the roots are real and distinct.

(iv) If $D>0$ but it is not a perfect square, then roots are irrational and unequal.

(v) If $D<0,$ then the roots are imaginary and are given by $\mathrm{\alpha }=\frac{-b+i\sqrt{4ac-b^2}}{2a}$ and $\mathrm{\beta }=\frac{-b-i\sqrt{4ac-b^2}}{2a}$.

(vii) If a quadratic equation in $x$ has more than two roots, then it is an identity in $x,$ that is $a=b=c=0.$

(viii) Surd root of an equation with "rational" coefficients always occur in pairs like $2+\sqrt{3}$ and $2-\sqrt{3}$. However, Surd roots of an equation with irrational coefficients may not occur in pairs.

3. Fundamental Theorem of Algebra:

(i) Every polynomial equation $f\left(x\right)=0$ has at least one root, real or imaginary (complex).

(ii) Every polynomial equation $f\left(x\right)=0$ of degree $n$ has exactly $n$ roots real or imaginary.

(iii) A polynomial equation of degree $n$ has exactly $n$ roots in $C$ when the roots are counted with their multiplicities.

4. Vieta’s Formula:

(i) Vieta’s Formula for Polynomial equation of degree $3$:

A cubic polynomial equation whose roots are $\alpha ,\beta$ and $\gamma$ is $x^3-(\alpha +\beta +\gamma )x^2+(\alpha \beta +\beta \gamma +\gamma \alpha )x-\alpha \beta \gamma =0$.

(ii) Vieta’s Formula for Polynomial equation of degree $\mathit{n}\mathbf{>}\mathbf{3}$:

If a monic polynomial equation of degree $n$ has roots ${\alpha }_1,{\alpha }_2,{\alpha }_3,...{\alpha }_n$, then 

5. A polynomial equation of degree $n$ with ${\alpha }_1,{\alpha }_2,{\alpha }_3.....{\alpha }_n$ is given by $x^n-\sum _{}^{}{\alpha }_1{x}^{n-1}+\sum _{}^{}{\alpha }_1{\alpha }_2{x}^{n-2}-\sum _{}^{}{\alpha }_1{\alpha }_2{\alpha }_3{x}^{n-3}+......+{\left(-1\right)}^n\sum _{}^{}{\alpha }_1{\alpha }_2{\alpha }_3.....{\alpha }_n{x}^n=0$.

6. If a complex number $z_0$ is a root of a polynomial equation with real coefficients, then its complex conjugate $\overline{z_0}$ is also a root.

7. Roots of Higher Degree Polynomial Equations:

A few results about polynomial equations that are useful in solving higher degree polynomial equations.

(i) Every polynomial in one variable is a continuous function from $R$ to $R$.

(ii) For a polynomial equation $P\left(x\right)=0$ of even degree, $P\left(x\right)\to \infty$ as $P\left(x\right)\to \pm \infty$. Thus the graph of an even degree polynomial start from left top and ends at right top. 

(iii) Every polynomial is differentiable any number of times.

(iv) The real roots of a polynomial equation $P\left(x\right)=0$ are the points on the $x-$axis where the graph of $P\left(x\right)=0$ cuts the $x-$axis.

(v) If $a$ and $b$ are two real numbers such that $P\left(a\right)$ and $P\left(b\right)$ are of opposite signs, then

(a) there is a point $c$ on the real line for which $P\left(c\right)=0$.

(b) that is, there is a root between $a$ and $b$.

(c) it is not necessary that there is only one root between such points; there may be $3,5,7,...$ roots; that is the number of real roots between $a$ and $b$ is odd and not even

8. Rational Root Theorem: 

Let $a_n{x}^n+.....+a_{1}x+a_0=0$ with $a_n,a_0\ne 0$ be a polynomial with integer coefficients. If $\frac{p}{q}$ with $\left(p,q\right)=1$, is a root of the polynomial, then $p$ is a factor of $a_0$ and $q$ is a factor of $a_n$.

9. Reciprocal Equations:

(i) A polynomial $P\left(x\right)$ of degree $n$ is said to be a reciprocal polynomial if one of the following conditions is true:

(a) $P\left(x\right)=x^{n}P\left(\frac{1}{x}\right)$

(b) $P\left(x\right)=-x^{n}P\left(\frac{1}{x}\right)$

(ii) A polynomial equation $a_n{x}^n+.....+a_{1}x+a_0=0; a_n\ne 0$ is a reciprocal equation if, and only if, one of the following two statements is true:

(a) $a_n=a_0, a_{n-1}=a_1, a_{n-2}=a_2......$

(b) $a_n=-a_0, a_{n-1}=-a_1, a_{n-2}=-a_2......$

(iii) A reciprocal equation cannot have $0$ as a solution.

(iv) The coefficients and the solutions are not restricted to be real.

(v) The statement “If $P\left(x\right)=0$ is a polynomial equation such that whenever $\alpha$ is a root, $\frac{1}{\alpha }$ is also a root, then the polynomial equation $P\left(x\right)=0$ must be a reciprocal equation” is not true.

10. Descartes Rule:

(i) A change of sign in the coefficients is said to occur at the $j^{\mathrm{th}}$ power of $x$ in a polynomial $P\left(x\right)$, if the coefficient of $x^{j+1}$ and the coefficient of $x^j$(or) also coefficient of $x^{j-1}$ coefficient of $x^j$ are of different signs. (For zero coefficient we take the sign of the immediately preceding nonzero coefficient).

(ii) If $p$ is the number of positive zeros of a polynomial $P\left(x\right)$ with real coefficients and $s$ is the number of sign changes in coefficients of $P\left(x\right)$, then $s-p$ is a nonnegative even integer.

(iii) The number of negative zeros of the polynomial $P\left(x\right)$ cannot be more than the number of sign changes in coefficients of $P\left(-x\right)$ and the difference between the number of sign changes in coefficients of $P\left(-x\right)$ and the number of negative zeros of the polynomial $P\left(x\right)$ is even.