Question

The remainder when f(x) = x⁴⁵ is divided by x² – 1 is ____________.

Answer 1

Let f(x) = x⁴⁵

To find the remainder obtained when x⁴⁵ is divided by x² – 1,

Let the remainder (r) be ax + b.

Then,
f(x) = (x² – 1) q + r
$$\begin{gathered} \Rightarrow x^{45}=\left(x^2-1\right)q+\left(ax+b\right)...\left(1\right) \\ \mathrm{Putting}x=1\mathrm{in}\mathrm{equation}\left(1\right),\mathrm{we}\mathrm{get} \\ {\left(1\right)}^{45}=\left({\left(1\right)}^2-1\right)q+\left(a\left(1\right)+b\right) \\ \Rightarrow 1=0+a+b \\ \Rightarrow a+b=1...\left(2\right) \\ \mathrm{Putting}x=-1\mathrm{in}\mathrm{equation}\left(1\right),\mathrm{we}\mathrm{get} \\ {\left(-1\right)}^{45}=\left({\left(-1\right)}^2-1\right)q+\left(a\left(-1\right)+b\right) \\ \Rightarrow -1=0-a+b \\ \Rightarrow -a+b=-1...\left(3\right) \\ \mathrm{Solving}\left(2\right)\mathrm{and}\left(3\right),\mathrm{we}\mathrm{get} \\ b=0\mathrm{and}a=1 \end{gathered}$$

Therefore, the remainder (r) = 1(x) + 0 = x.

Hence, the remainder when f(x) = x⁴⁵ is divided by x² – 1 is x.