A 5th degree ternary equation?

Algebra Level 5

Let f ( x ) = x 5 − 5 x + a f(x)=x^{5}-5x+a , a ∈ R a\in\mathbb R , Then which of the following statements are correct 1. I f a < − 4 t h e n 0 r e a l r o o t s . 2. I f a < − 4 t h e n 1 r e a l r o o t . 3. I f a < − 4 t h e n 2 r e a l r o o t s . 4. I f a < − 4 t h e n 3 r e a l r o o t s . 5. I f a < − 4 t h e n 4 r e a l r o o t s . 6. I f a < − 4 t h e n 5 r e a l r o o t s . 7. I f 4 > a > − 4 t h e n 0 r e a l r o o t s . 8. I f 4 > a > − 4 t h e n 1 r e a l r o o t . 9. I f 4 > a > − 4 t h e n 2 r e a l r o o t s . 10. I f 4 > a > − 4 t h e n 3 r e a l r o o t s . 11. I f 4 > a > − 4 t h e n 4 r e a l r o o t s . 12. I f 4 > a > − 4 t h e n 5 r e a l r o o t s . 13. I f a > 4 t h e n 0 r e a l r o o t s . 14. I f a > 4 t h e n 1 r e a l r o o t . 15. I f a > 4 t h e n 2 r e a l r o o t s . 16. I f a > 4 t h e n 3 r e a l r o o t s . 17. I f a > 4 t h e n 4 r e a l r o o t s . 18. I f a > 4 t h e n 5 r e a l r o o t s . S u b m i t t h e a n s w e r a s p r o d u c t o f t h e c o r r e c t o p t i o n s . \\ 1.\quad If\quad a<-4\quad then\quad 0\quad real\quad roots.\\ 2.\quad If\quad a<-4\quad then\quad 1\quad real\quad root.\\ 3.\quad If\quad a<-4\quad then\quad 2\quad real\quad roots.\\ 4.\quad If\quad a<-4\quad then\quad 3\quad real\quad roots.\\ 5.\quad If\quad a<-4\quad then\quad 4\quad real\quad roots.\\ 6.\quad If\quad a<-4\quad then\quad 5\quad real\quad roots.\\ 7.\quad If\quad 4>a>-4\quad then\quad 0\quad real\quad roots.\\ 8.\quad If\quad 4>a>-4\quad then\quad 1\quad real\quad root.\\ 9.\quad If\quad 4>a>-4\quad then\quad 2\quad real\quad roots.\\ 10.\quad If\quad 4>a>-4\quad then\quad 3\quad real\quad roots.\\ 11.\quad If\quad 4>a>-4\quad then\quad 4\quad real\quad roots.\\ 12.\quad If\quad 4>a>-4\quad then\quad 5\quad real\quad roots.\\ 13.\quad If\quad a>4\quad then\quad 0\quad real\quad roots.\\ 14.\quad If\quad a>4\quad then\quad 1\quad real\quad root.\\ 15.\quad If\quad a>4\quad then\quad 2\quad real\quad roots.\\ 16.\quad If\quad a>4\quad then\quad 3\quad real\quad roots.\\ 17.\quad If\quad a>4\quad then\quad 4\quad real\quad roots.\\ 18.\quad If\quad a>4\quad then\quad 5\quad real\quad roots.\\ Submit\quad the\quad answer\quad as\quad product\quad of\quad the\quad correct\quad options.


The answer is 280.

Prince Loomba by Prince Loomba , 21, India

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2 solutions

Let f ( x ) = x 5 − 5 x + a f(x)=x^5 - 5x + a

By descartes' rule of signs , we note that f ( x ) f(x) has atmost 3 distinct real roots for any value of a a .

Denote the points of isolated extremum as α 1 \alpha_1 and α 2 \alpha_2 . Note that f ′ ( α 1 ) = f ′ ( α 2 ) = 0 f'(\alpha_1)=f'(\alpha_2)=0 .

Now, f f has exactly 3 real distinct roots if and only if f ( α 1 ) f ( α 2 ) < 0 f(\alpha_1) f(\alpha_2) < 0

And f f has exactly one real (non repeated) root if and only if f ( α 1 ) f ( α 2 ) > 0 f(\alpha_1) f(\alpha_2) > 0

Now, f ( α 1 ) f ( α 2 ) = ( a − 4 ) ( a + 4 ) f(\alpha_1) f(\alpha_2) = (a-4)(a+4)

Hence the statements 2 , 10 2, 10 and 14 14 are right.

So, the answer is 280 \boxed{\boxed{280}}

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Prince Loomba
May 8, 2016

Let a ( x ) = x 5 − 5 x a n d b ( x ) = − a a(x)=x^{5}-5x\quad and\quad b(x)=-a . Draw the graph of a(x) to find out that for a<4, 1 solution, a>-4,1 solution and |a|<4, 3 solutions (By shifting the y axis).

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