Question

If (x-alpha) is a factor of f(x)=x^3-mx^2-2nax+nx^2 show that alpha=m+n and a not equal to zero

Answer 1

Q : If
$(x - a)$
is a factor of
${x}^{3} - m {x}^{2} - 2anx + n {x}^{2} = 0$
Ans :

Steps :
1)
$f(x) = {x}^{3} - m {x}^{2} - 2anx + n {x}^{2}$
If (x-a) is a factor of f(x), then f(a) =0

2)
$f(a) = {a}^{3} - m {a}^{2} - 2n {a}^{2} + n {a}^{2} \\ = > {a}^{3} - m {a}^{2} - n {a}^{2} \\ 0 = {a}^{2} (a - m - n) \\ = > a - m - n = 0 \\ = > a = m + n$
Since, a is not equal to zero.
Therefore,
a = m + n

Answer 2

To Prove ⇒ α = m + n

Proof ⇒

Given condition ⇒
 (x - α) is the factor of x³ - mx² - 2nαx + nx².

Using the Remainder Theorem, If (x - α) is the factor of the given polynomial, then the Remainder is equal to zero.

Now,
x - α = 0
x = α

Now,
f(α) = α³ - m(α)² -2nα² + n(α)²
0 = α³ - mα² - 2nα² + nα²
0 = α³ -mα² - nα²
α³ - α²(m + n) = 0
α³ = α²(m + n)
α =  m + n

Hence Proved.


Hope it helps.