Foundations — Remainder theorem and factor theorem — proof and applications

This page builds every single symbol used in the parent topic from absolute zero. If you already know polynomials cold, skim to the #Equipment checklist. Otherwise, read top to bottom — each idea is a brick for the next.

1. What is a variable $x$?

Think of a vending machine slot. You drop a number in, the machine does something with it, and out comes a result. The letter $x$ is that slot.

2. What is a polynomial $p(x)$?

For example: $p(x)=x^3-4x^2+6x-5$

Read the pieces:

• $x^3$ means $x\cdot x\cdot x$ — the small raised $3$ is the exponent, "how many copies of $x$ multiplied together".
• The numbers $-4$, $6$, $-5$ multiplying the powers are the coefficients.
• The lonely $-5$ (no $x$) is the constant term.
• The biggest exponent (here $3$) is the degree — the polynomial's "size rating".

Look at the figure: numbers flow into the box, the recipe fires, one number drops out. That is all $p(x)$ ever does.

3. Evaluating: the notation $p(a)$

Here $a$ is itself just a fixed number we haven't picked yet — a name for a chosen value, distinct from the free box $x$. When you meet $p(-3)$, you put $-3$ everywhere, including inside powers: $(-3{)}^2=9$, not $-9$. Watch signs carefully — this single habit prevents most beginner errors.

4. A picture of a polynomial: its graph and roots

In the figure, the curve crosses the flat axis at three spots. Each crossing is a root — a value of $x$ that makes the machine spit out exactly $0$. This "$=0$" moment is the star of the Factor Theorem, so keep the picture in mind: root = curve meets the axis.

5. Division with remainder — first for plain numbers

Before polynomials, remember ordinary division. Divide $17$ by $5$: $17=5\times 3+2$

6. Division with remainder — now for polynomials

The exact same shape of statement works for polynomials (this is Polynomial Long Division):

$p(x)=(x-a)\cdot q(x)+r$

Meet each symbol:

• $(x-a)$ is the divisor, a linear polynomial (degree $1$ — the highest power of $x$ is just $x^1$).
• $q(x)$ is the quotient polynomial — how many "copies" of $(x-a)$ fit.
• $r$ is the remainder.

The figure lines up the number statement $17=5\cdot 3+2$ against the polynomial statement so you can see they are the same idea wearing different clothes.

7. The symbol $(x-a)$ and its root

This is the pivot of the whole topic. Because $(x-a)=0$ at $x=a$, substituting $x=a$ into $p(x)=(x-a)q(x)+r$ makes the big messy $(x-a)q(x)$ term vanish, leaving $p(a)=r$. That single vanishing act is the entire proof of the Remainder Theorem.

8. The symbols $⟺$ and $\Rightarrow$

The Factor Theorem uses $⟺$: $(x-a)\text{ is a factor of }p(x)⟺p(a)=0$ This means "factor" and "$p(a)=0$" are the same news reported two ways. Prove one, you get the other free — but $⟺$ demands you check both directions, which the parent proof does.

9. Factor vs. "divides"

Prerequisite map

Everything on the left is built on this page; the two boxes on the right are the parent topic's twin results.

Where these lead next

Once these foundations click, the topic branches into Synthetic Division (a fast way to get $q(x)$), the Rational Root Theorem (which integers to test first), Factoring Polynomials and Solving Polynomial Equations (the payoff), all resting on the Fundamental Theorem of Algebra.

Equipment checklist

Read the question, answer aloud, then reveal.