Question

State the relationship between the zeros and the coefficient of a quadratic polynomial and cubic polynomial

Answer 1

The relationship between the zeros and the coefficient of polynomials is explained below.

Explanation:

• A polynomial is defined as an algebraic expression consisting of multiple terms. There are different types of polynomials such as linear, quadratic, cubic, etc.
• The general form of a quadratic polynomial is ax² + bx +c, where a ≠ 0.
• There are two zeroes for quadratic polynomial.  ( α and β )
• The relationship between the zeros and the coefficient of the quadratic polynomial is as follows:-

• Sum of zeroes= α+β =  $\frac{-b}{a}$ = $\frac{- \,Coefficient\,\ of\,\ x }{Coefficient\,\ of\,\ x^{2} }$      
• Product of zeroes =αβ  = $\frac{c}{a}$ = $\frac{Constant\,\ term }{Coefficient\,\ of\,\ x^{2} }$

• The general form of a cubic polynomial of ax³ + bx²+ cx + d, where a ≠ 0.
• There are three zeroes of a cubic polynomial.  (α, β, γ )

• The relationship between the zeros and the coefficient of the cubic polynomial is as follows:-

• Sum of zeroes = α+β+γ =  $\frac{-b}{a}$ = $\frac{- \,Coefficient\,\ of\,\ x^{2} }{Coefficient\,\ of\,\ x^{3} }$

• Sum of the product of zeroes =

                                 αβ+ βγ+ γα = $\frac{c}{a}$ =  $\frac{Coefficient\,\ of\,\ x }{Coefficient\,\ of\,\ x^{3} }$

• Product of zeroes= αβγ =  $\frac{-d}{a}$ = $\frac{-\, Constant\,\ term }{Coefficient\,\ of\,\ x^{3} }$