Question

if two zeroes of the polynomial x4-13x3+52x2-64x +24 are 3+√5 and 3-√5 find the other zero of this polynomial

Answer 1

Answer:

1 and 6

Step-by-step explanation:

Dividend = $x^4-13x^3+52x^2-64x +24$

Since we are given that  two of it's zeroes are $3+\sqrt{5}$ and  $3-\sqrt{5}$

So,$(x-3-\sqrt{5})(x-3+\sqrt{5}$

Using identity $(x+y)(x-y)=x^2-y^2$

$( x -3 )^2 - ( \sqrt{5} )^2$

$x^2+9-6x- 5$

$x^2-6x+4$

Now  Divisor = $x^2-6x+4$

Since we know that :

$Dividend =(Divisor \times Quotient)+Remainder$

$x^4-13x^3+52x^2-64x +24=(x^2-6x+4 \times x^2-7x+6)+0$

Now we will factorize the quotient

$x^2-7x+6=0$

$x^2-6x-x+6=0$

$x(x-6)-(x-6)=0$

$(x-1)(x-6)=0$

$x=1,6$

Hence the other zero of this polynomial is 1 and 6