Relationship between Zeroes and Coefficients of a Polynomials

A polynomial is an algebraic expression consisting of multiple terms. There are various types of polynomials such as linear, quadratic, cubic etc.  A real number k is a zero of a polynomial of p(x) if \(\displaystyle p\left( x \right)=0\)

Linear Polynomials

• The general form of linear polynomial is \(\displaystyle p\left( x \right)=ax+b;\,\,a\ne 0.\)

since,  \(\displaystyle p\left( x \right)=ax+b;\,\)

For zeroes, \(\displaystyle \begin{array}{l}p\left( x \right)=0\\ax+b=0\\ax=-b\\x=-\frac{b}{a}\end{array}\)

\(\displaystyle \alpha =-\frac{b}{a}=-\frac{{\text{Constant term}}}{{\text{Coefficient}\,\text{of}\,\text{x}}}\)

Quadratic Polynomials

General form of quadratic polynomial is \(\displaystyle p\left( x \right)=a{{x}^{2}}+bx+c;\,\,\,a\ne 0\)
There are two zeroes of a quadratic polynomial  i.e \(\displaystyle \alpha \,\,\And \,\,\beta \,\)

\(\displaystyle \text{sum of zeroes = }\alpha +\beta =-\frac{b}{a}=-\frac{{Coefficient\,of\,x}}{{Coefficient\,of\,{{x}^{2}}}}\)

\(\displaystyle \text{product of zeroes = }\alpha \beta =-\frac{c}{a}=-\frac{{\text{Constant term}}}{{Coefficient\,of\,{{x}^{2}}}}\)

Cubic Polynomials

General form of cubic polynomial is \(\displaystyle p\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d;\,\,a\ne 0\)                           There are two zeroes of a quadratic polynomial i.e \(\displaystyle \alpha \, \,\ \beta and \,\,\gamma \,\)

\(\displaystyle \text{Sum of zeroes =}-\frac{b}{a}=-\frac{{\text{Coefficient of }{{\text{x}}^{2}}}}{{\text{Coefficient of }{{\text{x}}^{3}}}}\)

\(\displaystyle Sum\text{ }of\text{ }the\text{ }product\text{ }of\text{ }zeroes\text{ }taken\text{ }two\text{ }at\text{ }a\text{ }time=-\frac{c}{a}=-\frac{{\text{Coefficient of x}}}{{\text{Coefficient of }{{\text{x}}^{3}}}}\)

\(\displaystyle \text{product of zeroes =}-\frac{d}{a}=-\frac{{\text{Constant term}}}{{\text{Coefficient of }{{\text{x}}^{3}}}}\)

Formation of Polynomials

Quadratic Polynomial:

If \(\displaystyle \alpha \,\,\And \,\,\beta \,\) are zeroes of quadratic polynomial then the quadratic polynomial will be-

\(\displaystyle \begin{array}{l}p\left( x \right)=k\left[ {{{x}^{2}}-\left( {\alpha +\beta } \right)x+\alpha \beta } \right]\\p\left( x \right)=k\left[ {{{x}^{2}}-\left( {sum\,of\,zeroes} \right)x+product\,of\,zeroes} \right]\end{array}\)

Cubic Polynomials

If \(\displaystyle \alpha ,\,\,\beta \,\,\And \,\,\gamma \)  are zeroes of cubic polynomial then the cubic polynomial will be-

\(\displaystyle p\left( x \right)=k\left[ {{{x}^{3}}-\left( {\alpha +\beta +\gamma } \right){{x}^{2}}+\left( {\alpha \beta +\beta \gamma +\gamma \alpha } \right)x-\alpha \beta \gamma } \right]\)