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+18 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. 1
  2. 2
  3. 3
  4. 4
asked in Set Theory & Algebra by Veteran (68.8k points)
edited by | 1.1k views

3 Answers

+25 votes
Best answer

yes, option D is the correct answer. 

Here is how p(x) should look like:


Value of P(x) will be zwro at green circled (O) points,So they will be the roots of the polynomial P(x).
Hence, the minimum degree of P(x) will be 4.

answered by Veteran (13.8k points)
edited by
@ anurag

in this graph will not touch at y=0 then how did we calculate degree ... using graph (other than arjun sir solution )

The value  of x is a root of polynomial ,p(x), if at that value p(x) = 0

p(1) = 1 , p(2) = -1 , x is going from 1 to 2 , p(x) is going from 1 to -1 , it is clear somewhere it goes though 0 in between, for x in  [1,2 ] ,  minimum one root lies there

similarly minimum one root , in between [2,3],[3,4], [4,5].

[Note : here , here we don't have no such given value of x for which p(x) crosses though 0. ]

yes ,sir this is clear ... i was talking about link which i gave there is no p(x)=0 then how we will get point using graph ...i am not getting , minimum one root , in between [2,3],[3,4], [4,5]. :(

p(2) = -1 p(3) = 1 , -1 to 1 , go though 0 . so root lies in [2,3] . 

In that link, it is guess , polynomial is degree 1, p(x) = ax+b , and should satisfy p(0) = 5, p(1) = 4, p(2) = 9 and p(3) = 20 if it does not then guess it is of degree2 , p(x) = ax2+bx+c and should satisfy p(0) = 5, p(1) = 4, p(2) = 9 and p(3) = 20. . and so on .

p(2) = -1 p(3) = 1 , -1 to 1 , go though 0 . so root lies in [2,3] .   am getting this no issue problem is this   if i want to solve  this  p(0) = 5, p(1) = 4, p(2) = 9 and p(3) = 20  by  graph then i all are going in increasing order rt ... that we cant find degree using graph ?? 

I believe graphs can give intuition for too.

On plotting all those points it can be observed that we will get a bowl shaped plot and any equation of the form "y = ax + b" can not have a bowl shaped plot since y = ax + b represents a straight line whose slope is 'a' & y intercept is 'b'..

You can also check it by differentiating y = ax + b with respect to x, it will give 'a' which is a constant, but in bowl shaped plot our slope is varying

So definitely the polynomial should have the degree strictly greater than 1.

& from the graph of y = x^2(which also is a polynomial of degree 2) we can observe that it has a bowl shape.

This implies that there exists a polynomial(x^2) who is of degree 2 and whose plot is bowl shaped.

So our polynomial could have 2 as its minimum degree since it has a bowl shaped plot.



We can do by plotting graph and there is pattern of graphs of different degree, as polynomial of degree 2 will be parabolic. But for exact plotting large no of data should be given . Else it is not good to use it.
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

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


+5 votes
Minimum Degree is 4 as there are atleast 4 roots possible for this polynomial
answered by Active (1.2k points)
how can we say that at least 4 roots exist?please xplain
+2 votes
answered by Loyal (3.7k points)
By above method not able get the same difference

1    -1     1    -1      1

    -2     2     -2    2

         4      -4     4

              -8       8

@Dileep kumar M 6

Do one more time

-8   8


4th difference same.So, Degree = 4


Does this method fail in any special case or order?
This should be the best answer...

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