Question

letp(x) be a polynomial of degree 4 such rhat p(n) 120/n for n= 1,2,3,4,5determine the value of p(6)​

Answer 1

Given :- let p(x) be a polynomial of degree 4 such that p(n) 120/n for n= 1,2,3,4,5 . Determine the value of p(6) ?

Answer :-

we have, a polynomial p(x) of degree 4 .

given that,

→ p(n) = (120/n) .

so,

→ p(1) = (120/1) = 120

→ p(2) = (120/2) = 60

→ p(3) = (120/3) = 40

→ p(4) = (120/4) = 30

→ p(5) = (120/5) = 24

we get,

→ p(1) = 2p(2) = 3p(3) = 4p(4) = 5p(5) .

then,

→ x*p(x) - 120 = m*(x-1)(x-2)(x-3)(x-4)(x-5)

→ x*p(0) - 120 = m*(0-1)(0-2)(0-3)(0-4)(0-5)

→ (-120) = m * (-1) * (-2) * (-3) * (-4) * (-5)

→ (-120) = m * (-120)

→ m = 1

then, required polynomial,

→ xp(x) - 120 = (x-1)(x-2)(x-3)(x-4)(x-5)

therefore,

→ p(6) = 6p(6) - 120 = (6-1)(6-2)(6-3)(6-4)(6-5) => 6p(6) - 120 = 5 * 4 * 3 * 2 * 1 => 6p(6) - 120 = 120 => 6p(6) = 240 => p(6) = (240/6) = 40 (Ans.)

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Answer 2

SOLUTION

GIVEN

p(x) be a polynomial of degree 4 such that

$\displaystyle \sf{p(n) = \frac{120}{n} \: \: for \: n \: = 1,2,3,4,5}$

TO DETERMINE

The value of p(6)

EVALUATION

Here it is given that

$\displaystyle \sf{p(n) = \frac{120}{n} \: \: for \: n \: = 1,2,3,4,5}$

So p(1) = 120 , p(2) = 60 , p(3) = 40 , p(4) = 30 , p(5) = 24

Thus we get from above

p(1) = 2p(2) = 3p(3) = 4p(4) = 5p(5) = 120

Since p(x) be a polynomial of degree 4

So xp(x) - 120 is a polynomial of degree 5 with zeroes as 1 , 2 , 3 , 4 , 5

xp(x) - 120 = k(x-1)(x-2)(x-3)(x-4)(x-5) - - - (1)

Where k is a constant to be determined

Putting x = 0 in both sides we get

- 120 = - 120k

∴ k = 1

Thus we get the polynomial as

xp(x) - 120 = (x-1)(x-2)(x-3)(x-4)(x-5)

In order to calculate p(6) we put x = 6

We get

6p(6) - 120 = (6-1)(6-2)(6-3)(6-4)(6-5)

$\sf{ \implies \: 6p(6) - 120 = 120}$

$\sf{ \implies \: 6p(6) = 240}$

$\sf{ \implies \: p(6) = 40}$

FINAL ANSWER

Hence the required value p(6) = 40

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