Question

By actual division show that (x+2)is a factor of the polynomial p(x)=2x^4+x^3-14x^2-19x-6.verify it by using factor theorem...... please give right answer... irrelevant answers will be reported...
please note that find the answer with actual division.so I need only division part ​

Answer 1

Step-by-step explanation:

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).