Question

Without actual division , prove that 3 x
4 – 2x
3
-2x
2
-2x -5 is exactly divisible by 3x
2 – 2x -5

Answer 1

Answer:

To prove :

x

4

+2x

3

−2x

2

+2x−3 is exactly divisible by x

2

+2x−3

Proof :

Let, p(x)=x

4

+2x

3

−2x

2

+2x−3

And, g(x)=x

2

+2x−3

Then, g(x)=x

2

+2x−3

=x

2

+3x−x−3

=x(x+3)−(x+3)

=(x+3)(x−1)

Now, we check if g(x) is a factor of p(x) by using factor theorem.

∴ (x+3) and (x−1) divides p(x) ifp(−3) and p(1)=0

So,

p(−3)=(−3)

4

+2(−3)

3

−2(−3)

2

+2(−3)−3

=81−54−18−6−3=0

and,

p(1)=(1)

4

+2(1)

3

−2(1)

2

+2(1)−3

=1+2−2+2−3=0

Hence, p(x) is divisible by (x+3) and (x−1)

⇒p(x) is divisible by (x+3)(x−1)

⇒p(x) is divisible by g(x)

Hence proved