Question

If √5 and -√5 are two zeroes of the polynomial x3+3x2

-5x-15 then its third zero

is______.​

Answer 1

Step-by-step explanation:

Given:-

√5 and -√5 are two zeroes of the polynomial x^3+3x^2-5x-15 .

To find:-

Find the third zero of the given Polynomial?

Solution:-

Given that

Given cubic polynomial P(x) = x^3+3x^2-5x-15 .

Given zeroes = √5 and -√5

We know that

By factor theorem

If √5 is the zero then (x-√5) is a factor of P(x)

If -√5 is the zero then (x+√5) is a factor of P(x)

=> (x-√5)(x+√5) is also a factor of P(x)

=>x^2 -5 is the factor of P(x)

Now to get another zero of P(x) then we have to divide P(x) by x^2-5.

(x^3+3x^2-5x-15)÷(x^2-5)

=> [x^2(x+3)-5(x+3)]÷(x^2-5)

=>[(x^2-5)(x+3)]÷(x^2-5)

On cancelling (x^2-5) then

=> x+3

=> x+3 is a factor of P(x)

=>-3 is the zero of P(x)

Since x+3 = 0=>x=-3

Answer:-

The third zero of the given Polynomial is -3