V3
Obtain all other zeroes of 2x3 + x2 - 6x - 3, if
two of its zeroes are - 3and V3.
Answers
Step-by-step explanation:
Zeros of the given polynomial are -\frac{1}{2},\sqrt3,-\sqrt3−
2
1
,
3
,−
3
Step-by-step explanation:
Two zeros of the polynomial 2x^3+x^2-6x-32x
3
+x
2
−6x−3 are\sqrt3,-\sqrt3
3
,−
3
If a and b are zeros of a polynomial then (x-a)(x-b) must be the factor.
Therefore, the factors are
(x-\sqrt3)(x+\sqrt3)(x−
3
)(x+
3
)
On applying the difference of squares formula (a+b)(a-b)=a^2-b^2(a+b)(a−b)=a
2
−b
2
\begin{gathered}(x-\sqrt3)(x+\sqrt3)\\\\=x^2-(\sqrt3)^2\\\\x^2-3\end{gathered}
(x−
3
)(x+
3
)
=x
2
−(
3
)
2
x
2
−3
Now, for other zero, we divide the given polynomial by x^2-3
\frac{2x^3+x^2-6x-3}{x^2-3}=2x+1
x
2
−3
2x
3
+x
2
−6x−3
=2x+1
Therefore, we have
\begin{gathered}2x+1=0\\\\x=-\frac{1}{2}\end{gathered}
2x+1=0
x=−
2
1
Therefore, zeros of the given polynomial are -\frac{1}{2},\sqrt3,-\sqrt3−
2
1
,
3
,−
3
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