Obtain all other zeros of x4 + 4x3
– x
2
– 10x + 6, if two of its zeros are – (1±√3)
Answers
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-15p(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})(x−
5
) and (x+
5
) are facotrs of given polynomial.
(x-\sqrt{5})(x+\sqrt{5})=x^2-5(x−
5
)(x+
5
)=x
2
−5
Divide the given polynomial p(x) by x^2-5x
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
x
2
−5
x
4
+4x
3
−2x
2
−20x−15
=x
2
+4x+3
Equate the quotient equal to 0, to find the remaining zeros.
x^2+4x+3=0x
2
+4x+3=0
x^2+3x+x+3=0x
2
+3x+x+3=0
x(x+3)+1(x+3)=0x(x+3)+1(x+3)=0
(x+3)(x+1)=0(x+3)(x+1)=0
Equate each factor equal to 0.
x=-3x=−3
x=-1x=−1
Therefore the other zeros of the given polynomial are -1 and -3.