Question

let p(x) be a fourth degree polynomial with coefficient of leading term 1 and p(1)=p(2)=p(3)=0 then find p(-1)+p(-5)/p(0)+p(4) ​

Answer 1

Given :   p(x) be a fourth degree polynomial with coefficient of leading term 1 and p(1)=p(2)=p(3)=0

To find : p(-1)+p(5)/p(0)+p(4) ​

Solution:

Let say

P(x)  = (x - 1)(x - 2)(x - 3)(x  - a)      as  coefficient of leading term 1 and p(1)=p(2)=p(3)=0

P(-1)  + p(5) =   (-1 - 1)(-1 - 2)(-1 - 3)(-1  - a)  +   (5 - 1)(5 - 2)(-5 - 3)(5  - a)

= (-2)(-3)(-4)(-a - 1)     + (4)(3)(2)(5 - a)

= 24a + 24  + 120 - 24a

= 144

p(0) + p(4) =   (0 - 1)(0 - 2)(0 - 3)(0  - a)  +   (4 - 1)(4 - 2)(4 - 3)(4  - a)

= (-1)(-2)(-3)(-a)     + (3)(2)(1)(4 - a)

= 6a + 24 - 6a

= 24

( p(-1)+p(5) ) / (p(0)+p(4) ​ )  = 144/24    =  6

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