Question

divide using synthetic division.
$(x^{2}+1)\div\left(x+\frac{1}{4}\
ight)=\square$

Explanation:

Step1: Identify root of divisor

For $x+\frac{1}{4}=0$, root is $x=-\frac{1}{4}$

Step2: List coefficients of dividend

Dividend $x^2+1$ has coefficients $1, 0, 1$

Step3: Perform synthetic division

Bring down 1. Multiply by $-\frac{1}{4}$: $1\times(-\frac{1}{4})=-\frac{1}{4}$
Add to next coefficient: $0+(-\frac{1}{4})=-\frac{1}{4}$
Multiply $-\frac{1}{4}$ by $-\frac{1}{4}$: $\frac{1}{16}$
Add to last coefficient: $1+\frac{1}{16}=\frac{17}{16}$

Step4: Write quotient and remainder

Quotient: $x-\frac{1}{4}$, remainder: $\frac{17}{16}$

Answer:

$x-\frac{1}{4}+\frac{\frac{17}{16}}{x+\frac{1}{4}}$ or $x-\frac{1}{4}+\frac{17}{16x+4}$