Question

find the zeroes of the quadrayic polynomial and verify the relationship between the zeroes and the co efficient:- x^2+7x+10
​

Answer 1

Given Polynomial : x² + 7x + 10

We've to find relationship b/w zeroes and Coefficient of given quadratic Polynomial.

⠀⠀⠀⠀

☆ Let's find out zeroes of Given Polynomial :

⠀⠀⠀⠀

$\begin{gathered}\qquad:\implies\sf x^2 +7x + 10 = 0\\\\\\ \qquad:\implies\sf x^2 + 2x + 5x + 10 = 0\\\\\\ \qquad:\implies\sf x(x + 2) +5(x + 2)= 0\\\\\\ \qquad:\implies\sf (x + 2)(x + 5) = 0\\\\\\ \qquad:\implies{\underline{\boxed{\pmb{\frak{\red{x = - 2\:\:or\:\:x = - 5}}}}}}\:\bigstar\\\\\\\end{gathered}$

$\therefore\:{\underline{\sf{Hence,\:The\:zeroes\:of\:Polynomial\:are\:{\pmb{ \sf{ -2}}}\:{\sf{\&}}\:{\bf{- 5}}.}}}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀

✇ Let's consider α and β be zeroes of Polynomial.

Here, In the given Polynomial x² + 7x + 10, {a = 1 , b = 7 & c = 10}.

⠀⠀⠀⠀

☆ Now, Let's verify the relationship between zeroes and Coefficient :

⠀⠀⠀

$\begin{gathered}\bf{\dag}\:{\underline{\boxed{\pmb{\sf{Sum\:of\:zeroes\:\purple{(\alpha + \beta)}\::}}}}}\\\\\\\end{gathered}$

$\begin{gathered}\qquad\quad\dashrightarrow\sf \alpha + \beta = \dfrac{-b}{a}\\\\\\ \qquad\quad\dashrightarrow\sf \bigg( - 2\bigg) + \bigg( - 5\bigg) = \dfrac{-7}{1}\\\\\\ \qquad\quad\dashrightarrow\sf - 2 - 5= - 7\\\\\\ \qquad\quad\dashrightarrow{\boxed{\boxed{\frak{\pink{ - 7 = - 7}}}}}\\\\\\\end{gathered}$

$\begin{gathered}\bf{\dag}\:{\underline{\boxed{\pmb{\sf{Product\:of\:zeroes\:\purple{(\alpha \beta)}\::}}}}}\\\\\\\end{gathered}$

$\begin{gathered}\qquad\quad\dashrightarrow\sf \alpha \beta = \dfrac{c}{a}\\\\\\ \qquad\quad\dashrightarrow\sf \bigg( - 2\bigg) \bigg( - 5\bigg) = \dfrac{10}{1}\\\\\\ \qquad\quad\dashrightarrow\sf 5 \times 2 = 10\\\\\\ \qquad\quad\dashrightarrow{\boxed{\boxed{\frak{\pink{10= 10}}}}}\\\\\\\end{gathered}$

Hence Verified !