Question

write relationship between zeros and co-coefficient of a quadratic polynomial​

Answer 1

Step-by-step explanation:

General form of quadratic polynomial is ax 2 + bx +c where a ≠ 0. There are two zeroes of quadratic polynomial. Product of zeroes =ca = Constant term Coefficient of x2 Constant term Coefficient of x 2 .

Answer 2

Answer:

Relationship between the Zeros and Coefficients of a Quadratic Polynomial:-

Let α and β be the zeros of a quadratic polynomial f(x) = ax² + bx + c. By factor theorem (x - α) and (x - β) are the factors of f(x).

∴        f(x) = k (x - α) (x - β), where k is a constant.

⇒       ax² + bx + c = k {x² - (α + β)x + αβ}

⇒       ax² + bx + c = kx² - k (α + β)x + kαβ

Comparing the coefficients of x², x, and constant terms on both sides, we get:-

        a = k, b = -k (α + β) and c = kαβ

⇒  α + β = $\sf{-\frac{b}{a}}$ and αβ = $\sf{\frac{c}{a}}$

⇒  α + β = $\sf{-\frac{Coefficient\:of\:x}{Coefficient\:of\:x^{2}}}$ and, αβ = $\sf{\frac{Constant\: term}{Coefficient\:of\:x^{2}}}$

Hence,

Sum of zeros = $\sf{-\frac{b}{a}=-\frac{Coefficient\:of\:x}{Coefficient\:of\:x^{2}}}$, Product of zeros = $\sf{\frac{c}{a}=\frac{Constant\:term}{Coefficient\:of\:x^{2}}}$

Remark If α and β are the zeros of a quadratic polynomial f(x). Then, the polynomial f(x) is given by

      f(x) = k {x² - (α + β)x + αβ}

or,  f(x) = k {x² - (Sum of zeros) x + Product of the zeros}