Question

obtain all the zeroes of the polynomial
${x}^{4} + 4 {x}^{3} - 2 {x}^{2} - 20x - 15$
if two of its zeroes are
$\sqrt{5} \\ \sqrt{ - 5}$
​

Answer 1

Given p(x) = x^4 + 4x^3 - 2x^2 - 20x - 15.

Given that \sqrt{5}, -\sqrt{5}

5

,−

5

are the zeroes.

Then,

(x + \sqrt{5})(x - \sqrt{5})(x+

5

)(x−

5

) is also a factor.

= > x^2 - (\sqrt{5})^2=>x

2

−(

5

)

2

=> x^2 - 5=>x

2

−5

------------------------------------------------------------------------------------------------------------

Now,

Divide the given polynomial by x^2 - 5.

x^2 - 5) x^4 + 4x^3 - 2x^2 - 20x - 15 ( x^2 + 4x + 3

x^4 - 5x^2

--------------------------------------------

4x^3 + 3x^2 - 20x - 15

4x^3 - 20x

----------------------------------------------

3x^2 - 15

3x^2 - 15

------------------------------------------------

0

---------------------------------------------------

Now,

We factorize x^2 + 4x + 3

= > x^2 + x + 3x + 3

= > x(x + 1) + 3(x + 1)

= > (x + 1)(x + 3)

= > (x + 1)(x + 3) = 0

= > x = -1,-3 is a zero of p(x).