Question

Without actual division prove that the
polynomial g(x) = x+a is a factor of
polynomial f (x)=x^5+a^5
​please answer fast

Answer 1

Answer:

ANSWER

Let p(x)=x

4

−4x

2

+12x−9 and g(x)=x

2

+2x−3

g(x)=(x+3)(x−1) Hence, (x+3) and (x−1) are factors of g(x).

In order to prove that p(x) is exactly divisible by g(x), it is sufficient to prove that p(x) is exactly divisible by (x+3) and (x−1).

∴ Let us show that (x+3) and (x−1) are factors of p(x).

Now, p(x)=x

4

−4x

2

+12x−9

p(−3)=(−3)

4

−4(−3)

2

+12(−3)−9=81−36−36−9=81−81=0

∴p(−3)=0

p(1)=(1)

4

−4(1)

2

+12(1)−9=1−4+12−9=13−13=0

∴p(1)=0

∴(x+3) and (x−1) are factors of p(x)⇒g(x)=(x+3)(x−1) is also fa factor of p(x).

Hence, p(x) is exactly divisible by g(x). i.e., (x

4

−4x

2

+12x−9) is exactly divisible by (x

2

+2x−3).