Question

if x-a is a factor of the polynomial x3-mx2-2nax+ na2.prove that a=m+n

Answer 1

using factors theorem

x = a

x^3 - mx^2 - 2nax + na^2 = 0

( a )^3 - m ( a )^2 - 2 na (a) + na^2 = 0

a^3 - ma^2 - 2na^2 + na^2 = 0

a^3 - ma^2 - na^2 = 0

a^2 ( a - m - n ) = 0

a^2 = 0. or. a - m - n = 0

a = m + n

Answer 2

Step-by-step explanation :

Given, (x - a) is a factor of the polynomial f(x) = x3 - mx2-2nax+ na2

By factor theorem, If (x-p) is a factor of f(x), then f(p) = 0.

In this case,

f(a) = 0

$a^3-ma ^ 2-2na(a)+ na^{2} = 0 \\ \\ a^3 - m {a}^{2} - 2n { a}^{2} + na^{2} = 0 \\ \\ a^3 - m{a}^{2} -n {a}^{2} = 0 \\ \\ ( a - m - n) a^2 0=$

So, Either a² = 0 or a - m - n = 0

a - m - n = 0

- m - n = - a

- ( m + n) = - a

a = m + n

Hence proved!