Question

If $\sqrt{5} and -\sqrt{5}$ are two zeroes of the polynomial $x^{3}+3x^{2}-5x-15$, then its third zero is
(a) 3
(b) -3
(c) 5
(d) -5

Answer 1

SOLUTION :

The correct option is (b) : - 3 .

Let α = √5 , β = - √5 and γ are the three Zeroes of the cubic  polynomial.

Given :  The cubic  polynomial f(x) = x³ + 3x² - 5x - 15

On comparing with  ax³ + bx² + cx + d  

a = 1, b= 3, c = - 5 , d = - 15

Sum of zeroes of cubic  polynomial= −coefficient of x² / coefficient of x³

α + β + γ = −b/a

√5 +(- √5) +  γ = −3/1

√5 - √5 +  γ = −3

γ = −3

Hence, the third zero (γ) is − 3 .

HOPE THIS ANSWER WILL HELP YOU..

Answer 2

Answer: x = -3

Solution:

if
$\sqrt{5} \: \: and \: \: - \sqrt{5}$
are the zeros of given polynomial,than

$(x + \sqrt{5} )(x - \sqrt{5} )$
are the factors of polynomial,on multiply or on applying identity
$(a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ (x - \sqrt{5} )(x + \sqrt{5}) = {x}^{2} - 5$
now divide the polynomial by this

${x}^{2} - 5) {x}^{3} + 3 {x}^{2} - 5x - 15(x + 3 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: {x}^{3} \: \: \: \: \: \: \: \: \: \: \: \: - 5x \\ - - - - - - - - - - - \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 3 {x}^{2} - 15 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 3 {x}^{2} - 15 \\ - - - - - - - - - - \\\: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \:0$
so third factor is
$x + 3$
third zero is
$x = - 3$