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

Answer»

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.


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