Question

given that
$x - \sqrt{5}$
is a factor of the polynomial
$x^{3} - 3 \sqrt{5} x ^{2 } - 5x + 15 \sqrt{5}$
,find all the zeros of the polynomial

Answer 1

Hiii friend,

(X-✓5) is a factor of the given polynomial.

G(X) = X-✓5

P(X) = X³-3✓5X²-5X+15✓5

Divide the P(X) by G(X) we get,

X-✓5)X³-3✓5X²-5X+15✓5(X²-2√5X+15

**†****X³-✓5X²
******+X³+✓5X²
******-****-
******________________

***********-2✓5X²-5X+15✓5
***********-2✓5X²+10X
************+******* -
________________
****************** -15X+15✓5
*******************-15X +15✓5
*******************+******-
******************__________
********************* 00

We get,

Quotient = X²-2✓5X+15

and,

Reminder = 0

P(X) = X²-2✓5X+15

Here,

A= 1 , B = -2✓5 and C = 15

X = (-B(+-)✓B²-4AC/2A

X = (2✓5(+-)(2✓5)²-12/2

X = ✓5 and +-✓2

HENCE,

✓5 , +-✓2 ARW THE TWO OTHER ZERSO OF THE GIVEN POLYNOMIAL.

HOPE IT WILL HELP YOU...... :-)