Question

Obtain all the other zeros of x4 + 4x3 - 2x2-20x - 15, if two of its zeros are root 5 and - root 5.

Answer 1

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Answer 2

Answer:

The other zeros of the given polynomial are -1 and -3.

Step-by-step explanation:

The given polynomial is

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

It is given that √5 and -√5 are two of its zeros. It means $(x-\sqrt{5})\text{ and }(x+\sqrt{5})$ are facotrs of given polynomial.

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

Divide the given polynomial p(x) by $x^2-5$, to find the remaining factors.

Using long division we get

$\frac{x^4+4x^3-2x^2-20x-15}{x^2-5}=x^2+4x+3$

Equate the quotient equal to 0, to find the remaining zeros.

$x^2+4x+3=0$

$x^2+3x+x+3=0$

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

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

Equate each factor equal to 0.

$x=-3$

$x=-1$

Therefore the other zeros of the given polynomial are -1 and -3.