For a linear divisor or factor $ax\pm b$, assume $a\ne0$. Given a polynomial $ \mathrm{f}(x)$ is divided by a linear expression $ax - b$, then its remainder is $ \mathrm{f} \left( \frac{b}{a} \right) $ This follows from the division, where $ \mathrm{f}(x) = (ax - b) \mathrm{Q}(x) + \mathrm{R}$ where $ \mathrm{Q}(x)$ is the quotient and $\mathrm{R}$ is the remainder $ \mathrm{f} \ \left(\frac{b}{a} \right) = \mathrm{Q}\left(\frac{b}{a}\right) \times \left(a \times \frac{b}{a} - b \right) + \mathrm{R} $ $ \mathrm{f} \left( \frac{b}{a} \right) = \mathrm{Q} \left( \frac{b}{a} \right) \times 0 + \mathrm{R} = \mathrm{R} $ Remainder when divided by higher degree polynomial When $ \mathrm{f}(x) $ is divided by a polynomial of degree $n$, its remainder is zero or has degree less than $n$ For example when $ \mathrm{f}(x)$ is divided by a quadratic expression $ax^2 + bx + c$, its remainder has the form $dx+e$, which may be linear, constant or zero. In summary: Divisor Remainder $ax + b$ degree $1$ Constant $ax^2 + bx + c$ degree $2$ Linear expression $dx + e$ or constant or zero $ax^3 + bx^2 + cx + d$ degree $3$ Quadratic expression $ex^2 + fx + g$ or lower degree, or zero The displayed degree of each divisor assumes its leading coefficient is nonzero. The zero polynomial has no degree in this convention.

Remainder Theorem

TGM Original Questions

For a linear divisor or factor $ax\pm b$, assume $a\ne0$.

Given a polynomial $ \mathrm{f}(x)$ is divided by a linear expression $ax - b$, then its remainder is $ \mathrm{f} \left( \frac{b}{a} \right) $
This follows from the division, where
$ \mathrm{f}(x) = (ax - b) \mathrm{Q}(x) + \mathrm{R}$
where $ \mathrm{Q}(x)$ is the quotient and $\mathrm{R}$ is the remainder

$ \mathrm{f} \ \left(\frac{b}{a} \right) = \mathrm{Q}\left(\frac{b}{a}\right) \times \left(a \times \frac{b}{a} - b \right) + \mathrm{R} $
$ \mathrm{f} \left( \frac{b}{a} \right) = \mathrm{Q} \left( \frac{b}{a} \right) \times 0 + \mathrm{R} = \mathrm{R} $

Remainder when divided by higher degree polynomial

When $ \mathrm{f}(x) $ is divided by a polynomial of degree $n$, its remainder is zero or has degree less than $n$
For example when $ \mathrm{f}(x)$ is divided by a quadratic expression $ax^2 + bx + c$, its remainder has the form $dx+e$, which may be linear, constant or zero.
In summary:

Divisor	Remainder
$ax + b$ degree $1$	Constant
$ax^2 + bx + c$ degree $2$	Linear expression $dx + e$ or constant or zero
$ax^3 + bx^2 + cx + d$ degree $3$	Quadratic expression $ex^2 + fx + g$ or lower degree, or zero

The displayed degree of each divisor assumes its leading coefficient is nonzero. The zero polynomial has no degree in this convention.

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Divisor	Remainder
\(ax + b\) degree \(1\)	Constant
\(ax^2 + bx + c\) degree \(2\)	Linear expression \(dx + e\) or constant or zero
\(ax^3 + bx^2 + cx + d\) degree \(3\)	Quadratic expression \(ex^2 + fx + g\) or lower degree, or zero