Question

obtain all zeros of X raise to power 4 + 4 x cube minus 2 x square - 20 x minus 15 if two of its zeros are root 5 and minus root 5

Answer 1

Hey!!!!

I'll give a proper answer on paper for best understanding

The zeros of the polynomial are √5,-√5 , -1 and -3

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Answer 2

Answer:

The remaining zeros of the polynomial are -3 and -1.

Step-by-step explanation:

The given polynomial is

$P(x)=x^4+4x^3-2x^2-20x-15$

It is given that $\sqrt{5}$ and $-\sqrt{5}$ are zeros of the given polynomial. it means $x-\sqrt{5}$ and $x+\sqrt{5}$ are the factors of given polynomial.

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

Use long division method to divide P(x) by $x^2-5$.

The quotient is

$Q(x)=x^2+4x+3$

$P(x)=(x^2-5)(x^2+3x+x+3)$

$P(x)=(x^2-5)(x(x+3)+(x+3))$

$P(x)=(x^2-5)(x+3)(x+1)$

To find the zeros equate P(x)=0.

$(x^2-5)(x+3)(x+1)=0$

$x^2-5=0\Rightarrow x=\pm \sqrt{5}$

$x+3=0\Rightarrow x=-3$

$x+1=0\Rightarrow x=-1$

Therefore the remaining zeros of the polynomial are -3 and -1.