find the LCM:a3+1+2a2+2a, a3-1 and a4+a2+1
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Find the LCM of a
2
−3a+2,a
3
−a
2
−4a+4,a(a
3
−8)
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Answer
We know that LCM is the least common multiple.
Factorise a
2
−3a+2 as follows:
a
2
−3a+2=a
2
−2a−a+2=a(a−2)−1(a−2)=(a−1)(a−2)
Now, factorise a
3
−a
2
−4a+4 as follows:
a
3
−a
2
−4a+4=a
2
(a−1)−4(a−1)=(a
2
−4)(a−1)=(a
2
−2
2
)(a−1)=(a+2)(a−2)(a−1)
(using identity a
2
−b
2
=(a+b)(a−b))
Finally, factorise a(a
3
−8) as follows:
a(a
3
−8)=a(a
3
−2
3
)=a(a−2)(a
2
+2a+2
2
)=a(a−2)(a
2
+2a+4)
(using identity a
3
−b
3
=(a−b)(a
2
+b
2
+ab))
Therefore, the least common multiple between the polynomials a
2
−3a+2, a
3
−a
2
−4a+4 and a(a
3
−8) is:
LCM=a×(a−1)×(a−2)×(a+2)×(a
2
+2a+4)=a(a−1)(a+2)[(a−2)(a
2
+2a+4)]
=a(a−1)(a+2)(a
3
−8) (using identity a
3
−b
3
=(a−b)(a
2
+b
2
+ab))
Hence, the LCM is a(a−1)(a+2)(a
3
−8).
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