Given that √2 is a zero of the cubic polynomial
6x3 + √2x2 -10x – 4√2, find its other two zeroes
Answers
Since, it is given that
2
and −
2
are the zeroes of the polynomial p(x)=2x
4
−3x
3
−3x
2
+6x−2, therefore, (x−
2
) and (x+
2
) are also the zeroes of the given polynomial. Now, consider the product of zeroes as follows:
(x−
2
)(x+
2
)
=(x)
2
−(
2
)
2
(∵a
2
−b
2
=(a+b)(a−b))
=x
2
−2
We now divide 2x
4
−3x
3
−3x
2
+6x−2 by (x
2
−2) as shown in the above image:
From the division, we observe that the quotient is 2x
2
−3x+1 and the remainder is 0.
Now, we factorize the quotient 2x
2
−3x+1 by equating it to 0 to find the other zeroes of the given polynomial:
2x
2
−3x+1=0
⇒2x
2
−2x−x+1=0
⇒2x(x−1)−1(x−1)=0
⇒(2x−1)(x−1)=0
⇒(2x−1)=0,(x−1)=0
⇒x=
2
1
,x=1
Hence, the other two zeroes of p(x)=2x
4
−3x
3
−3x
2
+6x−2 are
2
1
,1