Question

if f(x) is a monic polynomial of a least degree such that f(1)=5 f(2)=4 f(3)=3 f(4)=2 f(5)=1 ,then find the value of f(0)=?​

Answer 1

f(0) = 0 is the Answer

Step-by-step explanation:

Given:

f(1) = 1

f(2) = 2

f(3) = 3

f(4) = 16

To find : f(5)  and f(0)

Assume another function g(x) such that g(x) = f(x) - x

Hence,

g(1) = 0 = x-1 ..... (1)

g(2) = 0 = x-2 ...... (2)

g(3) = 0 = x-3 ...... (3)

From (1,2,3), we can say that

g(x) = k (x-1)(x-2)(x-3) ......  (4)

f(4) = 16 .... Given

g(4) = 16 - 4 = 12

Therefore, g(4) = k(4-1)(4-2)(4-3)

12= k(4-1)(4-2)(4-3)

K=2

Equation 4 becomes

g(x) = 2 (x-1)(x-2)(x-3)

g(5) = 2(5-1)(5-2)(5-3)

g(5) = $2 \times 4 \times 3 \times 2$

g(5) = 48

But we know that

g(x) = f(x) - x

g(5) = f(5) - 5

f(5) = 48 + 5

Thus, f(5) = 53 and f(0) = 0