Question

029. Find the quadratic polynomial whose zeroes are -2 and 5. Verify the relationship between zeroes and coefficients
the polynomial​

Answer 1

GIVEN :-

• Zeroes of quadratic polynomial are -2 and 5.

TO FIND :-

• The quadratic polynomial.

SOLUTION :-

As we know that , the quadratic polynomial is given by ,

$: \implies \displaystyle \sf \: polynomial = x ^{2} - ( \alpha + \beta )x + ( \alpha \beta )$

• ɑ = -2.
• β = 5.

$\\ : \implies \displaystyle \sf \: polynomial =x ^{2} - ( - 2 + 5)x + ( - 2 \times 5)$

$: \implies \displaystyle \sf \: polynomial =x ^{2} - 3x + ( - 10)$

$: \implies \underline{ \boxed{ \displaystyle \sf \: \bold{polynomial =x ^{2} - 3x - 10}}}$

____________________…

Sum of zeroes,

$\\ : \implies \displaystyle \sf \: \alpha + \beta = \frac{ - coefficient \: of \: x}{coefficient \: of \: x ^{2} }$

$: \implies \displaystyle \sf - 2 + 5 = \frac{ - ( - 3)}{1}$

$: \implies \underline{ \boxed{\displaystyle \sf \bold{3 = 3}}}$

Product of zeroes,

$\\ : \implies \displaystyle \sf \alpha \beta = \frac{constant \: term}{coefficient \: of \: x ^{2} }$

$: \implies \displaystyle \sf - 2 \times 5 = \frac{ - 10}{1}$

$: \implies \underline{ \boxed{ \displaystyle \sf \bold{ - 10 = - 10}}}$

Hence Verified.

Answer 2

$\underline{\underline{ \mathbb{QUADRATIC \: POLYNOMIAL:-}}}$

$\rm Let \: \alpha = - 2 \: and \: \beta = 5$

$\boxed{ \bf{Polynomial = x² - ( \alpha + \beta )x+( \alpha \beta )}}$

$→\sf~x² - [( - 2) + 5 ]x+ [( - 2) \times 5 )]$

$→\sf~{x}^{2} - ( 3)x + ( - 10)$

$→\sf~\underline{ \underline{ {x}^{2} - 3x - 10}}$

$\underline{\underline{ \mathbb {RELATION:-}}}$

$\rm In \: polynomial \: ax²+bx+c:-$

$\star~ \boxed{\bf{ Sum \: of \: polynomial = \dfrac{-b}{a}}}$

$\star~\boxed{ \bf {Product ~of~ polynomial = \dfrac{c}{a}}}$