Question

Find all zeros of the polynomial $2x^{4}+7x^{3}-19x^{2}-14x+30$ 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) = 2x⁴ + 7x³  - 19x²  - 14x + 30

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) =2x⁴ + 7x³  - 19x²  - 14x + 30 by g(x) = x² - 2

[DIVISION IS IN THE ATTACHMENT.]

Hence , all the zeroes of the given Polynomial are: (√2), (- √2), 3/2 & - 5  

HOPE THIS ANSWER WILL HELP YOU …..

Answer 2

Answer:

Polynomial : 2x

4

+7x

3

−19x

2

−14x+30

2

and −

2

are the zeroes of the polynomial

⇒x

2

−2 divides the given polynomial

x

2

−2)

2x

4

+7x

3

−19x

2

−14x+30

(2x

2

+7x−15

2x

4

−4x

2

−+

________________________________________

7x

3

−15x

2

−14x+30

7x

3

−14x

−+

_________________________________________

−15x

2

+30

−15x

2

+30

+−

_________________________________________

x

2x

4

+7x

3

−19x

2

−14x+30=(x

2

−2)(2x

2

+7x−15)

Solving, 2x

2

+7x−15=0 we get other two roots given polynomial.

x=

2a

−b±

b

2

−4ac

x=

4

−7±

49+120

=

4

−7±

169

x=

4

−7+13

,

4

−7−13

x=

2

3

,−5

All zeroes of polynomial are :

2

,−

2

,3/2,−5