Question

Find all the zeros of the polynomial $x^{3}+ 3x^{2}-2x-6$,if two of its zeros are $-\sqrt{2} and \sqrt{2}$

Answer 1

Method of finding the remaining zeros of a polynomial when some of its zeros are given:

We firstly write the factor of polynomial using given zeros and multiply them to get g(x). Then divide a given polynomial by g(x).

The quotient so obtained give other zeros of given polynomial and we factorise it to get other zeros.

SOLUTION:

Let f(x) = x³  + 3x²  - 2x - 6

Given : Two Zeroes of the polynomial f(x) are - √2 & √2. Therefore , (x + √2) & (x - √2) are the two factors of given Polynomial f(x).

(x + √2) (x - √2)   = x² - (√2)²

= x² - 2

[(a+b)(a - b) = a² - b² ]

x² - 2 is a factor of given Polynomial f(x)

Now, Divide f(x) = x³  + 3x²  - 2x - 6 by g(x) = x² - 2

[DIVISION IS IN THE ATTACHMENT.]

Hence , all the zeroes of the given Polynomial are: (√2), (- √2), -3 .

HOPE THIS ANSWER WILL HELP YOU …..

Answer 2

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.

Hope it helps!