Let us recall the concepts in Polynomial:
A Polynomial is an expression consisting of variables and coefficients which involves operations such as addition, multiplication and non-negative integers.
1. Polynomials of degrees \(1\), \(2\) and \(3\) are called linear, quadratic and cubic polynomials, respectively. A quadratic polynomial in \(x\) with real coefficients is of the form \(ax^2 + bx + c\), where \(a\), \(b\), \(c\) are real numbers with a \(a\neq 0\). A quadratic polynomial can have at most \(2\) zeroes and a cubic polynomial can have at most \(3\) zeroes.
2. The zeroes of a polynomial \(p(x)\) are precisely the \(x\)-coordinates of the points, where the graph of \(y = p(x)\) intersects the \(x\) -axis.
3. If \(\alpha \) is the zero of the linear polynomial \(p(x)=ax+b\) then
\(\alpha = \frac{-b}{a}\) \(= \frac{-\text{constant term}}{\text{coefficient of }x} \)
4. If \(\alpha, \beta\) is the zero of the quadratic polynomial \(p(x)=ax^2+bx+c\) then
\(\alpha+\beta = \frac{-b}{a} = \frac{-\text{coefficient of }x}{\text{coefficient of }x^2}\)
\(\alpha\beta = \frac{c}{a} = \frac{\text{constant term}}{\text{coefficient of }x^2}\)
The general form of the quadratic polynomial based on its zeros is given by
\(p(x)=x^2-(\text{sum of zeroes})x+\text{product of zeroes}\)
\(p(x)=x^2-(\alpha+\beta)x+\alpha\beta\).
5. If \(\alpha,\beta,\gamma\) are the zeros of the cubic polynomial \(p(x)=ax^3+bx^2+cx+d\) then
\(\alpha+\beta+\gamma = \frac{-b}{a} = \frac{-\text{coefficient of }x^2}{\text{coefficient of }x^3}\)
\(\alpha\beta+\beta\gamma+\gamma\alpha = \frac{c}{a} = \frac{\text{coefficient of }x}{\text{coefficient of }x^3}\)
\(\alpha\beta\gamma = \frac{-d}{a} = \frac{-\text{constant term}}{\text{coefficent of }x^3}\)
The general form of the cubic polynomial based on its zeros is given by
\(p(x)=x^3-(\text{sum of zeroes})x^2+(\text{taking two roots at a time})x-\text{product of zeroes}\)
\(p(x)=x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\beta\gamma+\gamma\alpha)x- \alpha\beta\gamma\).