Question

find the quadratic polynomial, whose zeroes are -2 and -5.verify the reaction ship between zeroes and coefficients of the polynomial
pls help me​

Answer 1

Given : Zeroes of the polynomial are -2 & -5.

We've to find the Quadratic polynonial whose zeroes are -2 and -5.

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☆ Let's find out the quadratic polynomial :

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

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

$\begin{gathered}\qquad:\implies\sf - 2 + - 5\\ \\ \\ \implies \sf \pink{ \frak{- 7}} \\ \\ \\ \end{gathered}$

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

$\begin{gathered}\qquad:\implies\sf ( - 2 ) ( - 5) \\ \\ \\ \implies \sf \pink{ \frak{10} }\\ \\ \\ \end{gathered}$

$\begin{gathered}\bf{\dag}\:{\underline{\boxed{\pmb{\sf{Quadratic \: polynomial = {x}^{2} - Sum \: of \: zeroes (x) + Product \: of \: zeroes\::}}}}} \\ \\ \\ \end{gathered}$

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

$\therefore\:{\underline{\sf{Hence,\:The\: quadratic\: polynomial \: is \:{\pmb{{x}^{2} + 7x + 10}}.}}}$

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

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

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

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

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

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

Hence Verified !