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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. $1$
2. $2$
3. $3$
4. $4$

Go through these first :

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 $4$.

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 :

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

Do by this method  ,  this is more easy .

by

easiest

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

@Rajesh Panwar

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

### 1 comment

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

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 $D$

by

Any reference/reason for why this behavior is as it is ?
Seems like some property it follows

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