Question

Q13 Find all the zeroes of the polynomial p(x) = x4 – 7x3 + 9x2 + 13x – 4, if two of its zeroes are 2 +V3
and
$2 - \sqrt{3}$
​

Answer 1

Answer: Other zeroes 4,-1

Step-by-step explanation:

We have,

Polynomial

$p(x) = x^4-7x^3+9x^2+13x-4$

Let a, b be the other roots of the given polynomial.

We know,  

Sum of all roots = -(-7)

=> a+b+(2-√3)+(2+√3)=7

=>a+b=3

Also,  

Product of roots = -4

=>a*b*(2+√3)*(2-√3)=-4

=>a*b=-4

Now,

Using above equations, we get .

$a-4/a=3\\ \\=>a^2-4=3a\\ \\=>a^2-3a-4=0\\ \\=>(a-4)(a+1)=0\\ \\=>a=4,-1$

Hence,  a=4,b=-1 or vice-versa.

Hence, other zeroes are 4,-1.