Question

If f(x) = x4 + ax3 + bx2 + cx + d is a polynomial such that f(1) = 5, f(2) = 10, f(3) = 15, f(4) = 20, then the value of f(5) + f(–5) is a)2027 b)3102 c) 3048 d)4012

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

If f(x) = x⁴ + ax³ + bx² + cx + d is a polynomial such that f(1) = 5 , f(2) = 10, f(3) = 15, f(4) = 20, then the value of

f(5) + f(–5) is

a) 2027

b) 3102

c) 3048

d) 4012

EVALUATION

Here it is given that

f(1) = 5, f(2) = 10, f(3) = 15, f(4) = 20

Let g(x) = f(x) - 5x

Then g(1) = 0 , g(2) = 0 , g(3) = 0 , g(4) = 0

Thus 1 , 2 , 3 , 4 are zeroes of g(x)

Without any loss of generality we assume that

$\sf{g(x) = a(x - 1)(x - 2)(x - 3)(x - 4)}$

$\sf{ \implies \: f(x) - 5x = a(x - 1)(x - 2)(x - 3)(x - 4)}$

$\sf{ \implies \: f(x) = a(x - 1)(x - 2)(x - 3)(x - 4) + 5x}$

Here f(x) = x⁴ + ax³ + bx² + cx + d

Since f(x) is a polynomial with leading coefficient as 1

Thus we get a = 1

Above gives

$\sf{ \implies \: f(x) = (x - 1)(x - 2)(x - 3)(x - 4) + 5x}$

- - - - - - - Equation 1

Putting x = 5 we get

$\sf{ \implies \: f(5) = (5 - 1)(5- 2)(5- 3)(5 - 4) + (5 \times 5)}$

$\sf{ \implies \: f(5) =24 + 25}$

$\sf{ \implies \: f(5) =49}$

Putting x = - 5 in Equation 1 we get

$\sf{ \implies \: f( - 5) = ( - 5 - 1)( - 5- 2)( - 5- 3)( - 5 - 4) + ( - 5 \times 5)}$

$\sf{ \implies \: f( - 5) =3024 - 25}$

$\sf{ \implies \: f( - 5) =2999}$

Thus we get

$\sf{f(5) + f( - 5) = 49 + 2999 = 3048}$

FINAL ANSWER

Hence the correct option is c) 3048

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. If 1/2 is a root of the quadratic equation x²-mx-5/4=0 , then the value of m is?

https://brainly.in/question/21062028

2.Find the roots of the quadratic equation 6x2 +5x + 1 =0 by method of completing the squares.

https://brainly.in/question/30446963

3. Write a quadratic equation whose root are -4 and -5

https://brainly.in/question/24154410

Answer 2

Given :  f(x) = x4 + ax3 + bx2 + cx + d is a polynomial such that f(1) = 5, f(2) = 10, f(3) = 15, f(4) = 20

To Find :  value of f(5) + f(–5)

a)2027 b)3102 c) 3048 d)4012

Solution:

f(x) = x⁴ + ax³ + bx² + cx + d

f(1)  =  1  + a + b + c + d = 5  => a + b +  c + d  = 4   Eq1

f(2) = 16 + 8a  + 4b + 2c + d  = 10 => 8a + 4b  + 2c  + d = - 6  Eq2

f(3) = 81 + 27a  + 9b +3c + d  = 15 => 27a + 9b  + 3c  + d = - 66  Eq3

f(4) = 256 + 64a  + 16b +4c + d  = 20 => 64a + 16b  + 4c  + d = - 236  Eq4

Eq2 - 2 * Eq1

=> 6a + 2b  -  d  = -14   Eq5

Eq3 - 3 * Eq1

=> 24a + 6b - 2d = -78

=> 12a + 3b  - d   = - 39   Eq6

Eq4  - 4 * Eq1

=> 60a + 12b -  3d  = -252

=> 20a + 4b -  d =  - 84   Eq7

Eq6 - Eq5   =>   6a  + b =  - 25

Eq7 - Eq 6 =>    8a  + b  = -  45

=> 2a  = - 20 => a  = - 10

=> b = 35

2 * Eq 5 - Eq6

=> b  -  d  = 11

=> d = b - 11 =  35 -  11  = 24

a + b +  c + d  = 4  

-10 + 35 + c + 24 = 4  =>  c = - 45

f(5)  = 5⁴ + a5³ + b5² + 5c  + d

f(-5)  = (-5)⁴ + a(-5)³ + b(-5)² - 5c  + d

f(5) + f(-5)  =  2(625  +  25b  +   d  )

b = 35  , d  =  24

= 2( 625 + 25(35) + 24)

= 3048

f(x) = x⁴ -10x³ + 35x² -45x + 24

Learn More:

Write the polynomial in variable 'x'whose zero is -k/x​ - Brainly.in

brainly.in/question/15193112

following is a zero of the polynomial p(x) = (x+5)(x-2 ...

brainly.in/question/23655105