Question

how to use remainder theorem for x^4 +1 divided by x+1 with division steps please... really urgent ...

Answer 1

Remainder theorem

When a polynomial f(x) is divided by x-$\alpha$ and the division is continued until the remainder is not smaller than divisor, then the remainder is f($\alpha$).

Here,

f(x)=x⁴+1

Divisor=x+1

Comparing with x-$\alpha$

we will get, $\alpha=-1$

$\sf{\therefore{Remainder=f(\alpha)=f(-1)}}$

$\sf{\therefore{Remainder=(-1)^{4}+1}}$

$\sf{\therefore{Remainder=1+1=2}}$

The remainder is 2.

_______________________________

Verification:

⠀⠀⠀⠀⠀x-1 ) x⁴+0x³+0x²+1 ) x³+x²+1

⠀⠀⠀⠀⠀ x⁴- x³

⠀⠀⠀⠀-⠀⠀+

⠀⠀⠀__________

⠀⠀⠀⠀⠀x³+0x²

⠀⠀⠀⠀⠀x³- x²

⠀⠀⠀⠀-⠀⠀+

⠀⠀⠀__________

⠀⠀⠀⠀⠀x²+1

⠀⠀⠀⠀⠀x²-1

⠀⠀⠀⠀-⠀⠀+

⠀⠀⠀__________

⠀⠀⠀⠀⠀⠀⠀2

Hence verified, since remainder is 2.