If a polynomial \(f(x)\) is divided by \((x-a)\), then the remainder is \(f(a)\).

When \(f(x)\) is divided by \((x-a)\), we can write

\[ f(x)=(x-a)Q(x)+R \]

where \(Q(x)\) is the quotient and \(R\) is the remainder.

Substitute \(x=a\):

\[ f(a)=(a-a)Q(a)+R \] \[ f(a)=0+R \] \[ f(a)=R \]

So the remainder is \(f(a)\).

Find the remainder when

\[ f(x)=2x^3-3x^2+4x-5 \]

is divided by \((x-2)\).

By the remainder theorem, the remainder is \(f(2)\):

\[ f(2)=2(2^3)-3(2^2)+4(2)-5 \] \[ =16-12+8-5 \] \[ =7 \]

Find the remainder when

\[ f(x)=x^4+2x^2-7 \]

is divided by \((x+1)\).

Since \((x+1)=(x-(-1))\), use \(a=-1\).

\[ f(-1)=(-1)^4+2(-1)^2-7 \] \[ =1+2-7 \] \[ =-4 \]

Find \(k\) if the remainder when

\[ f(x)=x^3+kx^2-2x+1 \]

is divided by \((x-1)\) is \(5\).

By the remainder theorem:

\[ f(1)=5 \]

\[ 1^3+k(1)^2-2(1)+1=5 \] \[ 1+k-2+1=5 \] \[ k=5 \]

The factor theorem is a special case of the remainder theorem.

If the remainder is \(0\), then \((x-a)\) is a factor.