Question

Find all zeroes of the polynomial f(x) x3 + 3x² – 2x – 6,
if two of its zeroes are

√2 and - √2.​

Answer 1

Given: We're provided with a polynomial f ( 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.

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━

⌬ 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.

⠀⠀⠀⠀⠀

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

⠀⠀⠀⌑ 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,

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

⠀

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

Answer 2

Given :-

x³ + 3x² - 2x - 6

To Find :-

All zeroes

Solution :-

Two zeroes are √2 and -√2

So

(x - √2)(x + √2) =0

(x × x) + (√2 × -√2) = 0

x² + (-2) = 0

Now By dividing

x² - 2 = 0

x² - 2) x³ + 3x²- 2x - 6 (x + 3

         x³          - 2x

      -            +    

                 + 3x² - 6

                  + 3x² - 6

________________________

                    0         0

Now

x + 3 = 0

x = 0 - 3

x = -3