Q7. Find all the zeroes of the polynomial x^4– 8x^3+ 19x² - 12x + 2
,if two of its zeroes are 2+√2 and 2-√2.
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Answer:
Let, p(x)=x
4
−x
3
−8x
2
+2x+12
Since,
2
and −
2
are zeros of p(x)
⇒(x−
2
) and (x+
2
) divides p(x) (∵Factor thm.)
⇒(x−
2
)(x+
2
) divides p(x)
⇒(x
2
−2) divides p(x)
x
2
−2)
x
4
−x
3
−8x
2
+2x+12
( x
2
−x−6
−
x
4
+
−
2x
2
−x
3
−6x
2
+2x+12
+
−
x
3
−
+
2x
−6x
2
+12
+
−
6x
2
−
+
12
0
Now, the other zeros can be obtained from on solving x
2
−x−6=0
⇒x
2
−3x+2x−6=0
⇒x(x−3)+2(x−3)=0
⇒(x−3)(x+2)=0
⇒x=−2,3
Hence, all the zeros are −2,3,
2
,−
2
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