Question

prove that the polynomial f(x)=x^2n +a^2n, n € r\{0}is not divisible by g(x) = x+a​

Answer 1

Answer:

Step-by-step explanation:

We will solve this by using principle of Mathematical Induction:

First, take n=1

We get that:

x^2+a^2

We can see that it is not divisible by x+a.

Therefore, the given polynomial is not divisible by x+a.

Answer 2

Concept

The remainder theorem of polynomial p(x) says that if a polynomial is divided by (x-a), then the remainder obtained is p(a). It is an approach of Euclidean division of polynomials.

Given

a polynomial f(x)=x^2n +a^2n, n € r\{0}

Find

we are asked to prove that f(x) is not divisible by g(x) = x+a​

Solution

if f(x) is divisible by (x+a)​,

then f(-a) must be equal to 0.

Thus, substituting -a in the function, we get

f(-a)= (-a)^2n +a^2n

f(-a)= a^2n +a^2n

f(-a)= 2a^2n

This value can never be equal to 0 unless a= 0.

Thus, the polynomial f(x)=x^2n +a^2n, n € r\{0} is not divisible by g(x) = x+a​ unless a = 0 itself.

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