How do you factor completely h3−27h+10?

1 Answer
Shwetank Mauria
May 9, 2017

h3−27h+10=(h−5)(h2+5h−2)=(h−5)(h+52+√332)(h+52−√332)

Explanation:

As the function is h3−27h+10, one of the zeros could be factor of 10 i.e. ±2 or ±5. As is seen 5 is a zero of the function as

(5)3−27(5)+10=125−135+10=0

and hence (h−5) is a factor of h3−27h+10

Dividing it by (h−5), we get

h2(h−5)+5h(h−5)−2(h−5)=(h−5)(h2+5h−2)

As discriminant of h2+5h−2 is 52−4×1×(−2)=33, which is not a perfect square and hence we can only have additional irrational factor.

h2+5h−2=(h2+2×52×h+(52)2)−(52)2−2

= (h+52)2−334=(h+52)2−(√332)2

= (h+52+√332)(h+52−√332)

Hence h3−27h+10=(h−5)(h+52+√332)(h+52−√332)

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