For a polynomial P and linear divisor ax+b, a 0: Remainder=P!(- ba) Set the divisor equal to zero to find the value to substitute. In the examples below, f(x)=2x3-x+3. Examples **Divisor** | **Remainder** | **Example: f(x)=2x3-x+3** x-c | f(c) | Divided by x-2: f(2)=16-2+3 =17 ax-b | f!(ba) | Divided by 2x+1: f!(- 12)=- 14+ 12+3 =3 14

For a polynomial \(P\) and linear divisor \(ax+b\), \(a\ne0\):

\(\text{Remainder}=\mathrm{P}\!\left(-\dfrac ba\right)\)

Set the divisor equal to zero to find the value to substitute.

In the examples below, \(\mathrm{f}(x)=2x^3-x+3\).

Divided by \(x-2\):
\(\begin{aligned} \mathrm{f}(2)&=16-2+3 \\ &=17 \end{aligned}\)

\(\mathrm{f}\!\left(\dfrac{b}{a}\right)\)

Divided by \(2x+1\):
\(\begin{aligned} \mathrm{f}\!\left(-\dfrac12\right)&=-\dfrac14+\dfrac12+3 \\ &=3\dfrac14 \end{aligned}\)

Remainder Theorem

Remainder Theorem

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