Question

2) Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and product of its zeroes as 3, -1 and -3 respectively.​

Answer 1

Given that , Sum of the zeroes ( α+β+γ ) is 3 , Sum of the product of it's zeroes taken two at at a time ( or , Sum of the Product of it's zeroes ) ( αβ+βγ+γα ) is – 1 & , Product of it's zeroes ( αβγ ) is – 3 .

⠀⠀⠀⠀⠀¤ We have to find a ❝ Cubic Polynomial  ❞ .

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━

❍ Let's say that, three zeroes of Cubic Polynomial be α , β & γ⠀, respectively.

$\qquad \qquad \maltese\: \underline {\pmb{\mathbb{ CUBIC \: POLYNOMIAL \:\::\:\: } }\:}$

As, We know that ,

$\qquad\underline {\boxed {\pmb{ \:\maltese \;Sum \:of \:zeroes \:\:\red {\:( \:\alpha \: + \beta \:+\:\gamma )}\::}}}$

${\qquad \dashrightarrow \sf \bigg( \alpha \:+ \beta \:+\:\gamma \bigg)}$

⠀⠀⠀⠀⠀⠀$\bigstar \underline {\boldsymbol{\:Now \: By \: Substituting \: the \: given \: Values \::}}$

$\qquad \dashrightarrow \sf \bigg( \alpha +\:\beta \: + \:\gamma \bigg) \:=\:3 \:\\\\ \dashrightarrow \underline{\boxed{\purple{\pmb{\frak{ \:\alpha +\:\beta \: + \:\gamma \:=\: 3 \: }}}}}\:\:\bigstar$

⠀⠀⠀⠀⠀AND ,

$\qquad\underline {\boxed {\pmb{ \:\maltese \;Sum \:of \:the \:products \:of \:it's \:zeroes \:\:\red {\:( \:\alpha\beta \: + \beta\gamma \:+\:\gamma\alpha )}\::}}}$

${\qquad \dashrightarrow \sf \bigg( \:\alpha\beta \: + \beta\gamma \:+\:\gamma\alpha \bigg)}$

⠀⠀⠀⠀⠀⠀$\bigstar \underline {\boldsymbol{\:Now \: By \: Substituting \: the \: given \: Values \::}}$

$\qquad \dashrightarrow \sf \bigg( \:\alpha\beta \: + \beta\gamma \:+\:\gamma\alpha \bigg) \:=\:-1 \:\\\\ \dashrightarrow \underline{\boxed{\purple{\pmb{\frak{ \:\:\alpha\beta \: + \beta\gamma \:+\:\gamma\alpha\:\:=\:\:-1 \: }}}}}\:\:\bigstar$

⠀⠀⠀⠀⠀AND ,

$\qquad\underline {\boxed {\pmb{ \:\maltese \;Product \:of \:zeroes \:\:\red {\:( \:\alpha \: \beta \:\:\gamma )}\::}}}$

${\qquad \dashrightarrow \sf \bigg( \alpha \: \beta \:\:\gamma \bigg)}$

⠀⠀⠀⠀⠀⠀$\bigstar \underline {\boldsymbol{\:Now \: By \: Substituting \: the \: given \: Values \::}}$

$\qquad \dashrightarrow \sf \bigg( \alpha \:\beta \: \:\gamma \bigg) \:=\:-3 \:\\\\ \dashrightarrow \underline{\boxed{\purple{\pmb{\frak{ \:\alpha \:\beta \: \:\gamma \:=\: -3 \: }}}}}\:\:\bigstar$

⠀⠀⠀⠀▪︎⠀We know , that if we have Sum of the zeroes , Sum of the product of it's zeroes taken two at at a time ( or , Sum of the Product of it's zeroes ) & , Product of it's zeroes and we have to find ❝ Cubic Polynomial ❞ then the used formula is given by :

$\qquad \star \:\:\underline {\boxed {\pmb{\sf Cubic _{(\:Polynomial \:)}\:=\: x^3 \:-\: \big\lgroup \sf{ \alpha \:+ \beta \:+\:\gamma }\big\rgroup\:x^2\: + \:\big\lgroup \sf{ \:\:\alpha\beta \: + \beta\gamma \:+\:\gamma\alpha\:\: }\big\rgroup\:x\:-\:\big\lgroup \sf{ \alpha \: \beta \:\:\gamma }\big\rgroup }}}$

$\qquad :\implies \sf x^3 \:-\: \big\lgroup \sf{ \alpha \:+ \beta \:+\:\gamma }\big\rgroup\:x^2\: + \:\big\lgroup \sf{ \:\:\alpha\beta \: + \beta\gamma \:+\:\gamma\alpha\:\: }\big\rgroup\:x\:-\:\big\lgroup \sf{ \alpha \: \beta \:\:\gamma }\big\rgroup$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad :\implies \sf x^3 \:-\:\big\lgroup \sf{ \alpha \:+ \beta \:+\:\gamma }\big\rgroup\:x^2\: + \:\big\lgroup \sf{ \:\:\alpha\beta \: + \beta\gamma \:+\:\gamma\alpha\:\: }\big\rgroup\:x\:-\:\big\lgroup \sf{ \alpha \: \beta \:\:\gamma }\big\rgroup \\\\\\ \qquad :\implies \sf x^3 \:-\:\big\lgroup \sf{ 3 }\big\rgroup\:x^2\: + \:\big\lgroup \sf{ \:\:-1\:\: }\big\rgroup\:x\:-\:\big\lgroup \sf{ -3 }\big\rgroup \\\\\\ \qquad :\implies \sf x^3 \:-\: 3 \:x^2\: \:-\:x\:+\: 3$

⠀⠀⠀⠀∴ Hence , The  Cubic Polynomial is ❝ x³ – 3x² – x + 3 ❞ .