By actual division, find the quotient and the remainder when

Question:

By actual division, find the quotient and the remainder when $\left(x^{4}+1\right)$ is divided by $(x-1)$. Verify that remainder $=f(1)$.

 

Solution:

Let $f(x)=x^{4}+1$ and $g(x)=x-1$

Quotient $=x^{3}+x^{2}+x+1$

Remainder $=2$

Verification:

Putting $x=1$ in $f(x)$, we get

$f(1)=1^{4}+1=1+1=2=$ Remainder, when $f(x)=x^{4}+1$ is divided by $g(x)=x-1$

 

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