Find the quadratic polynomial, the sum of whose zeros is 0 and their product is -1. Hence, find the zeros of the polynomial.

2 thoughts on “Find the quadratic polynomial, the sum of whose zeros is 0 and their product is -1. Hence, find the zeros of the polynomial.”

1. [tex]\bold AnswEr:-[/tex]

   Quadratic polynomial = x² – 1

   Zeroes of polynomial = -1 & 1

   [tex]\rule{200}{1}[/tex]

   Explanation:-

   Let the zeroes of polynomial be α & β

   Given:-

   • α + β = 0
   • αβ = -1
   • We know the standard form of a quadratic polynomial:-

   [tex]\star\: \boxed{\boxed{\sf\blue{x^2 – (Sum \: of \: zeros)x + Product\: of\: zeros }}}[/tex]

   Here,

   ⇒ Polynomial = x² – (0)x + (-1)

   ⇒ Polynomial = x² – 0x – 1

   ⇒ Polynomial = x² – 1

   Therefore,

   [tex]\therefore\underline{\textsf{Quadratic polynomial = {\textbf{x$^2$- 1}}}}[/tex]

   [tex]\rule{200}{1}[/tex]

   • We got the quadratic polynomial as x² – 1

   • Now we can find the zeros of polynomial by factorization method:-

   ⇒ x² – 1 = 0

   [We know, a² – b² = (a + b)(a – b) ]

   ⇒ (x + 1)(x – 1) = 0

   ⇒ x = -1 or x = 1

   Therefore,

   [tex]\therefore\underline{\textsf{Zeros of polynomial = {\textbf{-1 \& 1 }}}}[/tex]

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2. Step-by-step explanation:

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   [tex] \maltese \: \large \underline{ \bf{Answer:-}}[/tex]

   Given :

   [tex] \bold {• \: \: \: S um \:of \: \: zeroes \: \: of \: \: a \: \: quadratic \: \: polynomial \: \: ( \alpha + \beta ) = 0 }[/tex]

   [tex] \bold {• \: \: \:product \: \:of \: \: zeroes \: \: of \: \: a \: \: quadratic \: \: polynomial \: \: ( \alpha \beta ) = – 1 }[/tex]

   To Find :

   • Zeroes of the polynomials

   Solution :

   General form for quadratic equations,

   [tex] {\bf{ \boxed{ {x}^{2} – ( \alpha + \beta )x + \alpha \beta }}}[/tex]

   we have that ,

   [tex] \alpha + \beta = 0[/tex]

   [tex] \alpha \beta = – 1[/tex]

   Substitute the values in the form ,

   [tex] \bf \implies \: {x}^{2} – (0)x -1[/tex]

   [tex] \implies \bf \: {x}^{2} – 1[/tex]

   Next , find the zeroes of the polynomial.

   It is in the form (a²–b²)

   •°• (a²–b²) = (a+b)(a-b)

   [tex] \implies \sf \: {x}^{2} – {1}^{2}[/tex]

   [tex] \implies \sf \:(x + 1)(x – 1) = 0 [/tex]

   [tex]\implies \sf \:x = – 1 \: \: and \: \: 1[/tex]

   •°• Zeroes of the polynomial is ±1 !!

   _______________________

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