Question

find the zeros of the polynomial x3 + 3x² - 2x - 6 if two of its zeros are -√2 and + √2

Answer 1

Given:

• We're provided with a polynomial ( x ) : x³ + 3x² – 2x – 6 & if two of it's zeroes are (√2) and (– √2) respectively.

Need to find:

• We've to find out all zeroes of the given polynomial.

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⌬ Since, two zeroes of the given Polynomial x³ + 3x² – 2x – 6 are √2 and – √2.

Therefore,

➟ (x + √2) (x – √2)

➟ (x² – √2)²

➟ (x² – 2)

Here, x² – 2 is a factor of a given polynomial.

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✇ ${\underline{\mathcal{\pmb{\purple{BY\;USING\;DIVISION\;ALGORITHM :\::}}}}}$

⠀⠀⠀⌑ f( x ) = g( x ) × q( x ) - r( x ) ⌑

• f( x ) = x³ + 3x² – 2x – 6
• g( x ) = x² – 2
• q( x ) = x + 3
• r( x ) = 0

Therefore,

$\begin{gathered}\dashrightarrow\sf x^2 + 3x^2 - 2x - 6 = \Big\{x^2 - 2\Big\}\times \Big\{x + 3\Big\} - 0 \\\\\\\dashrightarrow\sf x^2 + 3x^2 - 2x - 6 = \Big\{x + \sqrt{2}\Big\}\Big\{x - \sqrt{2}\Big\} \Big\{x +3 \Big\} \\\\\\\dashrightarrow\underline{\boxed{\pmb{\frak{\purple{x = -\sqrt{2} + \sqrt{2}\;\&-3}}}}}\;\bigstar\end{gathered}$

⠀

∴ Hence, the required zeroes of the given polynomial are – √2, √2 and – 3 respectively.