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35 votes

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

- $1$
- $2$
- $3$
- $4$

2

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

2

51 votes

Best answer

1

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

3

6 votes

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 $D$

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

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