Question

If two zeroes of the polynomial f(x) = x3 + 3x2 2x 6 are --√2 and √2 find other zeroes

Answer 1

Answer:

-√2,√2,-3

Step-by-step explanation:

given, Two zeroes are -√2 and √2.

Sum of these zeroes = -√2 + √2 = 0

Product of these zeroes = (-√2)(√2) = -2.

∴ A quadratic polynomial with given zeroes is x² - 0x - 2 (or) x² - 2.

Since -√2 and √2 are zeroes of the given polynomial,so x² - 2 is a factor of given polynomial.

Dividing the given polynomial x³ + 3x³ - 2x - 6 by x² - 2, we get

x² - 2) x³ + 3x² - 2x - 6 ( x + 3

        x³           - 2x

       -----------------------

                 3x²  -  6

                 3x²  -  6

      -----------------------

                           0.

∴ By division algorithm,

x³ + 3x² - 2x - 6 = (x² - 2)(x + 3) + 0

                         = (x² - 2)(x + 3).

Quotient q(x) = x + 3.

Zeroes of q(x) are given by q(x) = 0.

x + 3 = 0

x = -3.

∴ Hence, all the zeroes of given polynomial are -√2,√2 and -3.

Answer 2

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.

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━

⌬ 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,

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

⠀

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