Question

If x = 2 and x = 0 are roots of the polynomial f(x) = 2x^3- 5x^2+ mx +n. Find the value of m and n.​

Answer 1

Step-by-step explanation:

◾x = 2 and x = 0 are the zeros of the polynomial f(x) = 2x^3- 5x^2+ mx +n.

◾First, put x = 2 in the given equation

f(x) = 2x^3- 5x^2+ mx +n.

f(2) = 2 ( 2)^3 - 5( 2 )^2 + m (2) + n

equate the equation with zero

↪ 2 ( 8 ) - 5 ( 4 ) + 2m + n = 0

↪ 16 - 20 + 2m + n = 0

↪ - 4 + 2m + n = 0

↪ -4 + 2m + n = 0 ...........(1)

◾Now, put x = 0 in the given equation,

f ( x ) = 2x^3- 5x^2+ mx +n.

f ( 0 ) = 2 ( 0 )^3 + 5 ( 0 )^2 + n

↪0 + 0 + n = 0

↪ n = 0

therefor the value of n = 0

Let us consider,n= 0 as a equation2

n = 0 ..............(2)

◾Compare both the equations (1) and (2)

-4 + 2m + n = 0 , n = 0

both the equations have,

RHS = RHS = 0

Therefor also LHS = LHS

↪-4 + 2m + n = n

↪-4 + 2m = n - n

↪-4 + 2m = 0

↪ 2m = 4

↪ m = 4 /2

↪ m = 2

◾Therefor Values of m and n are

$\boxed{\textbf{\large{m = 2 And n = 0 }}}$