ALG-09

Remainder theorem

P(r) is the remainder of P(x) divided by (x − r). Quadratic snapshot.

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PolynomialsRemainder theorem

Governing equation

P(r)=ar^2+br+c

$r \in \left[-10, 10\right]$

where

a: a (—)
b: b (—)
c: c (—)
r: r (—)
P(r): Remainder (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

P(r)=ar^2+br+c

$r \in \left[-10, 10\right]$

$r = -10$ $P(r) = 117$

Along the curve

②

Domain of the variable — blue

r_{\min} = -10,\quad r_{\max} = 10

$r \in \left[-10, 10\right]$

$r = -10$ $r = -10$

Along the curve

③

Response curve — green

P(r)=ar^2+br+c

$r \in \left[-10, 10\right]$

$r = -10$ $P(r) = 117$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}P(r)}{\mathrm{d}r}

$r \in \left[-10, 10\right]$

$r = -9.874$ $P(r)^{\prime} = -21.75$

Along the curve

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-09 — Remainder theorem) is the form associated with Remainder theorem. Working symbols: $a$ , $b$ , $c$ , $r$ $\rightarrow$ $P(r)$ . Write P(x) = (x−r) Q(x) + R. Setting x = r gives R = P(r).

Purpose

Purpose: compute

P(r)

from

a

b

c

r

in Algebra via

P(r)=ar^2+br+c

P(r) is the remainder of P(x) divided by (x − r). Quadratic snapshot. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 1.000\,\mathrm{—}

b = -2.000\,\mathrm{—}

c = -3.000\,\mathrm{—}

r = 2.000\,\mathrm{—}

, the governing relation

P(r)=ar^2+br+c

yields

P(r) = -3.0000\,\mathrm{—}

. P(x) = ax² + bx + c, evaluate at r. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a1 —b b-2 —c c-3 —r r2 —

Outputs

Remainder P(r)-3.0000 —

Reading speed

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Narration of this film

P(x) = ax² + bx + c, evaluate at r.

Write P(x) = (x−r) Q(x) + R. Setting x = r gives R = P(r).

Reading speed

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