GATE CSE 2000 | Question: 2.4

Question

A polynomial $p(x)$ satisfies the following:

• $p(1) = p(3) = p(5) = 1$ 
• $p(2) = p(4) = -1$

The minimum degree of such a polynomial is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Comments

Easier Method

https://math.stackexchange.com/questions/675110/what-is-the-minimum-degree-of-a-polynomial-given-the-initial-conditions/675137#675137

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Go through these first :

https://en.wikipedia.org/wiki/Critical_point_(mathematics)
https://socratic.org/questions/how-many-critical-points-can-a-function-have
Not mandatory but informative :
https://math.stackexchange.com/questions/1472888/critical-points-vs-inflection-points

Answer 1

yes, option $D$ is the correct answer. 

Here is how $p(x$) should look like:

Value of $P(x$) will be zero at circled $(O)$ points, so they will be the roots of the polynomial $P(x)$.
Hence, the minimum degree of $P(x)$ will be $4$.

Comments

ok yes ... you mean type of parabola ... not straight line means not linear rt ... ok clear , praveen sir also right ...  thank you ... praveen sir, anurag :

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i would like to add some important point here,

if a polynomial has minimum degree n then it will have atmost 'n' , x-intercepts and atmost 'n-1' turns. here in this graph we have 3 turns, we can say that it has minimum degree of 3+1 = 4.

sometimes we may get less than 'n' , x-intercepts, so we can check using number of turns in the graph.

example in this question https://gateoverflow.in/2245/gate1997-4-4

if we draw graph like above we will not get any x intercept but it has 1 turn, so we can say that it has minimum degree 1+1=2

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in order to solve this type of problem always go to this link it will never fail

https://math.stackexchange.com/questions/675110/what-is-the-minimum-degree-of-a-polynomial-given-the-initial-conditions/675137#675137

Answer 2

https://math.stackexchange.com/questions/675110/what-is-the-minimum-degree-of-a-polynomial-given-the-initial-conditions/675137#675137 

Do by this method  ,  this is more easy .

 answer : D 

Comments

easiest

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it means when we get all values same it will provide degree

@Rajesh Panwar

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yes

Answer 3

For a linear polynomial $p$, you'll always have $p(n+1)−p(n)$ the same. If you write down a table 
 

1     2    3     4   5
p(1) p(2) p(3) p(4) p(5)

which in our case would be this: 
 

1    2    3    4   5
1   -1    1   -1   1

 

and then write the differences $p(2)−p(1)$, $p(3)−p(2)$, etc in a row beneath, you'd get (again in our case) 
 

1    2    3    4   5
1   -1    1   -1   1
  -2    2   -2   2

That new row is called the "first differences". For a linear function, the entries would all be the same. You can also write down second, third and fourth differences: 
 

1    2    3    4   5
1   -1    1   -1   1
  -2    2   -2   2
     4   -4   4
       -8   8
          16
      
For a function with degree 2, the second differences will all be the same. 
In our case fourt difference is same. So degree is 4

Answer is $D$

Ref: https://math.stackexchange.com/questions/675110/what-is-the-minimum-degree-of-a-polynomial-given-the-initial-conditions/675137#675137

Comments

Any reference/reason for why this behavior is as it is ?

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Seems like some property it follows

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@Sachin Mittal 1 SIr by the method which you shared does only the magnitude have to be same or even the value??

Answer 4

Minimum Degree is 4 as there are atleast 4 roots possible for this polynomial

Comments

how can we say that at least 4 roots exist?please xplain