Question

On dividing
x^5-4x^3 + x² + 3x +1 by polynomial g(x)
the quotient and the remainder are x² - 1 and 2
respoctively find g(x)​

Answer 1

Let q(x),g(x) and r(x) be quotient, divisor and remainder respectively.

By remainder theorem,

f(x)=q(x)g(x)+r(x)

∴x5−4x3+x2+3x+1=(x2−1)g(x)+2

⇒(x2−1)g(x)=x5−4x3+x2+3x−1

Now,

x2−1)x5−4x3+x2+3x−1 ( x3−3x+1=g(x)

             −x5+−x3

             −3x3+x2+3x−1

             +−3x3−+3x

             x2−1

             −x .

Answer 2

Answer:

g(x)=x

3

−3x+1

Step-by-step explanation:

Let q(x),g(x) and r(x) be quotient, divisor and remainder respectively.

By remainder theorem,

f(x)=q(x)g(x)+r(x)

∴x

5

−4x

3

+x

2

+3x+1=(x

2

−1)g(x)+2

⇒(x

2

−1)g(x)=x

5

−4x

3

+x

2

+3x−1

Now,

x

2

−1)

x

5

−4x

3

+x

2

+3x−1

( x

3

−3x+1=g(x)

−

x

5

+

−

x

3

−3x

3

+x

2

+3x−1

+

−

3x

3

−

+

3x

x

2

−1

−

x

2

+

−

1

0

Hence, q(x)=x

3

−3x+1