Question

If two zeros of the polynomial f (x) = x^4 - 6x³ - 26x²+ 138x - 35 are 2 + √3 and 2 -√3 find others zeroes

Answer 1

Answer:

All the zeroes are 2 + √3,2 - √3,(-5),7

Step-by-step explanation:

Since 2+ √3 is a zero,

x-(2+√3) is a factor of the polynomial.

Since 2-√3 is a zero,

x-(2-√3) is a factor of the polynomial.

This means that (x-2-√3)(x-2+√3) are factors

(x-2-√3)(x-2+√3) [ using identity ]

= (x-2)² - (√3)²

= x² -4x +4 - 3

= x²- 4x+1

x²- 4x+1 is a factor of x⁴-6x³-26x²+138x-35

WE have to perform division algorithm ,

divide x⁴-6x³-26x²+138x-35 with x²- 4x+1

we get the answer as :-

x²- 2x - 35

Now split the middle terms !!

x²- 7x + 5x - 35

x [ x - 7 ] + 5 [ x - 7 ]

[ x+ 5 ] [ x - 7 ]

The other zeroes are :-

-5 and 7