Question

find the quadratic polynomial the sum of the sum and product of whose zeroes are 0 and root 2 respectively​

Answer 1

Answer:

$x^{2}$+√2

Step-by-step explanation:

Sum = 0

Product = root2

Polynomial = x^2-sx+P

= x^2-0+root2

=x^2+root2

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Answer 2

Given that , The sum and Product of zeroes are 0 & √2 , respectively.

Need To Find : The Quadratic Polynomial ?

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━

$\qquad \qquad \underline{\pmb{\mathbb{\bigstar \:\:QUADRATIC \:\:POLYNOMIAL \:\::\:}}}$

As , We know that,

$\qquad \underline {\boxed {\pmb{ \:\maltese \:Sum \:\: of \:\:zeroes \:\:\purple{ \:(\: \alpha + \beta \;)\:} \:: \: }}}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:+\:\beta \:\Big\} \:\: \\\\\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:+\:\beta \:\Big\}\:\:=\:\:\:\: 0 \\\\\dashrightarrow \underline {\boxed {\pmb{\pink{ \frak { \: \alpha \:+\:\beta \: \:\:=\:\:0 \:\:}}}}}\:\:\bigstar$

⠀AND ,

$\qquad \underline {\boxed {\pmb{ \:\maltese \:Product \:\: of \:\:zeroes \:\:\purple{ \:(\: \alpha \beta \;)\:} \:: \: }}}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:\:\beta \:\Big\} \:\: \\\\\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\\\\dashrightarrow \sf \Big\{ \: \alpha \:\:\beta \:\Big\}\:\:=\:\:\:\:\sqrt{2} \\\\ \dashrightarrow \underline {\boxed {\pmb{\pink{ \frak { \: \alpha \:\:\beta \: \:\:=\:\:\sqrt{2} \:\:}}}}}\:\:\bigstar$

Now , As , We know that ,

$\qquad \dag\:\:\bigg\lgroup \pmb{\sf \:Quadratic \:Polynomial\:\::\:x^2 \: - \:\{ \:Sum \:of \:zeroes \:\}x \:+ \:\{ \:Product \:of \:zeroes \} \: }\bigg\rgroup$

⠀⠀Here , Sum of Zeroes is α + β & Product of zeroes is α β

$\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: - \:\{ \:Sum \:of \:zeroes \:\}x \:+ \:\{ \:Product \:of \:zeroes \} \:$

$\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: - \:\{ \:\alpha + \beta \:\}x \:+ \:\{ \:\alpha \beta \} \: \\\\\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: - \:\{ \:0 \:\}x \:+ \:\{ \:\sqrt{2} \} \: \\\\\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: \:+ \:\{ \:\sqrt{2} \} \: \\\\\qquad \dashrightarrow \sf \:Quadratic \:Polynomial\:\:=\:x^2 \: \:+ \: \:\sqrt{2} \: \\\\\dashrightarrow \underline {\boxed {\pmb{\pink{ \frak { \: \:Quadratic \:Polynomial\:\:=\:x^2 \: \:+ \: \:\sqrt{2} \: \:\:}}}}}\:\:\bigstar$