ALG-10

Factor-theorem snapshot

(x − r) divides P iff P(r) = 0. The sheet reports P(r) as a zero-test.

Reading speed

PolynomialsFactor theorem

Governing equation

(x-r)\mid P\iff P(r)=0

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

where

a: a (—)
b: b (—)
c: c (—)
r: Trial root (—)
P(r): P(r) (0 ⇒ factor) (—)

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

(x-r)\mid P\iff P(r)=0

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

$r = -10$ $P(r) = 156$

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

(x-r)\mid P\iff P(r)=0

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

$r = -10$ $P(r) = 156$

Along the curve

④

Rate of change — amber

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

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

$r = -9.874$ $P(r)^{\prime} = -24.75$

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-10 — Factor-theorem snapshot) is the form associated with Factor theorem. Working symbols: $a$ , $b$ , $c$ , $r$ $\rightarrow$ $P(r)$ . A corollary of the remainder theorem. Synthetic division finds the quotient Q.

Purpose

Purpose: compute

P(r)

from

a

b

c

r

in Algebra via

(x-r)\mid P\iff P(r)=0

(x − r) divides P iff P(r) = 0. The sheet reports P(r) as a zero-test. 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 = -5.000\,\mathrm{—}

c = 6.000\,\mathrm{—}

r = 2.000\,\mathrm{—}

, the governing relation

(x-r)\mid P\iff P(r)=0

yields

P(r) = 0.0000\,\mathrm{—}

. Same quadratic as the remainder sheet. Near-zero P(r) means a numerical factor. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a1 —b b-5 —c c6 —Trial root r2 —

Outputs

P(r) (0 ⇒ factor) P(r)0.0000 —

Reading speed

Watch on YouTube

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Narration of this film

Same quadratic as the remainder sheet. Near-zero P(r) means a numerical factor.

A corollary of the remainder theorem. Synthetic division finds the quotient Q.

Reading speed

Watch on YouTube