Polynomial long division and synthetic division: Foundations

This page assumes you have seen nothing. Before we can even read the sentence "divide $2 x^{3} + 3 x^{2} - 5 x + 1$ by $x + 2$ ", we must know what each of those marks means and why it is there. We build every symbol from the ground up, each one leaning on the one before it. The parent note is Polynomial long division and synthetic division — come back here whenever a symbol there feels unearned.

[!definition] Symbol 0 — the letter $x$ (a "box for a number")

The very first mark we need is the letter $== x ==$ . It is not a mystery quantity. It is a labelled empty box that a number will later sit inside. When we write $x$ , we mean "some number, but I don't want to commit to which one yet."

Everything else on this page is built out of boxes and the four operations $+ - \times \div$ you already know from arithmetic.

[!definition] Symbol 1 — a power like $x^{3}$ (repeated multiplication)

A small raised number is an exponent. It counts how many copies of the box get multiplied together.

$x^{3} = x \times x \times x, x^{2} = x \times x, x^{1} = x, x^{0} = 1.$

The special case $x^{0} = 1$ is why a plain number like $7$ is secretly $7 x^{0}$ — it is a "power of size zero", the flat bar of height $1$ . We need this to know that a lone number is still part of the same family.

Why does the topic need powers? Because "how big is this piece of a polynomial" is answered entirely by its highest power — see degree below.

[!definition] Symbol 2 — a coefficient (the "how many" number)

In $2 x^{3}$ , the $2$ is the coefficient. It simply says how many copies of $x^{3}$ we have.

$2 x^{3} = x^{3} + x^{3} .$

Picture: if $x^{3}$ is one block, then $2 x^{3}$ is two identical blocks stacked. A coefficient can be negative ( $- 5 x$ means "take away five copies of $x$ ") or a fraction. A term with no visible number, like $x^{4}$ , secretly has coefficient $1$ : $x^{4} = 1 x^{4}$ . A missing term, like the $x^{2}$ that is absent from $x^{3} + 5$ , secretly has coefficient $0$ .

[!definition] Symbol 3 — a term, and combining "like terms"

A term is one coefficient glued to one power: $2 x^{3}$ , or $- 5 x$ , or the lone $1$ . Two terms are like terms when they have the same power. Only like terms can be added:

$3 x^{2} + 4 x^{2} = 7 x^{2}, but 3 x^{2} + 4 x stays 3 x^{2} + 4 x .$

[!definition] Symbol 4 — a polynomial $P (x)$

Chain several terms with $+$ and $-$ and you have a polynomial. We give it a name-box $P$ and write $== P (x) ==$ , read "P of x", meaning "the recipe $P$ that eats a number $x$ and gives back a number."

$P (x) = 2 x^{3} + 3 x^{2} - 5 x + 1.$

We write it in descending order — biggest power first, down to the lone number. See the figure: descending order lines the "bars" up tallest-to-shortest, which is exactly the order division attacks them.

Formally a polynomial is defined and named in the definition & degree note. The letters $D (x)$ , $Q (x)$ , $R (x)$ in the parent are just other name-boxes for other polynomials — Divisor, Quotient, Remainder.

[!definition] Symbol 5 — the degree $de g$

The degree of a polynomial is the highest power that actually appears (with a non-zero coefficient). We write it $de g (P)$ .

$de g (2 x^{3} + 3 x^{2} - 5 x + 1) = 3, de g (x + 2) = 1, de g (7) = 0.$

Fill-in reveal:

de g

of the constant

5

0

, because

5 = 5 x^{0}

and the highest power present is

0

[!definition] Symbol 6 — a linear divisor $x - c$

A linear polynomial has degree $1$ : its tallest bar is just $x^{1}$ . The special shape $== x - c ==$ (where $c$ is some fixed number) is the star of synthetic division.

$x - 2$ has $c = 2$ .
$x + 3$ is really $x - (- 3)$ , so $c = - 3$ .

The figure shows the straight-line graph of $y = x - c$ crossing zero at $x = c$ . Every case is covered:

divisor	rewrite as $x - c$	value $c$
$x - 5$	$x - (5)$	$+ 5$
$x + 4$	$x - (- 4)$	$- 4$
$x$	$x - (0)$	$0$

The last row is the degenerate case $c = 0$ : dividing by plain $x$ just shifts every power down by one — perfectly legal.

[!definition] Symbol 7 — the division statement itself

Now we can finally read the master equation the parent hangs everything on:

$P (x) = D (x) \cdot Q (x) + R (x), de g (R) < de g (D) or R = 0.$

In plain words: the thing we started with ( $P$ ) equals divisor times quotient, plus a small leftover ( $R$ ) that is too short to divide further. This mirrors whole numbers exactly:

$847 = 23 \times 36 + 19, 0 \leq 19 < 23.$

[!definition] Symbol 8 — the fraction bar as "divide"

When the parent writes $\frac{2 x ^{3}}{x} = 2 x^{2}$ , the bar means divide, and dividing powers just subtracts exponents:

$\frac{x ^{3}}{x ^{1}} = x^{3 - 1} = x^{2}, \frac{2 x ^{3}}{x} = 2 x^{2} .$

Why? $\frac{x ^{3}}{x} = \frac{x \cdot x \cdot x}{x}$ — one $x$ cancels top and bottom, leaving $x \cdot x = x^{2}$ . This single move is Step 1 of long division: "how many divisors fit the tallest bar?"

Where these feed the topic

The endpoint NEXT is where you go after this: the Factor & Remainder Theorem, which explains why a remainder of zero means $c$ is a root. From there the same division powers factoring polynomials, the rational root theorem, and partial fractions.

[!recall]- Explain each symbol to a 12-year-old

The letter $x$ is an empty box waiting for a number. A little raised number like the $3$ in $x^{3}$ says "multiply three copies of the box". The big number in front, like the $2$ in $2 x^{3}$ , says "how many of these". Stringing them together with plus and minus makes a polynomial — a to-do list of boxes. Its degree is just the size of its biggest box. And $x - c$ is a simple divisor that switches off exactly when $x$ equals $c$ , which is the only number synthetic division ever needs.

Equipment checklist

Read the left side, answer aloud, then reveal.

What does the exponent

3

x^{3}

literally instruct you to do?

Multiply three copies of

x

together:

x \times x \times x

What is

x^{0}

, and why does it make a lone number like

7

part of the polynomial family?

x^{0} = 1

, so

7 = 7 x^{0}

— a "size-zero" term.

What secret coefficient does a missing term (e.g. the absent

x^{2}

x^{3} + 5

) carry?

Coefficient

0

x^{3} + 0 x^{2} + 0 x + 5

. You must keep its column.

Which terms may be added together, and which may not?

Only like terms (same power).

3 x^{2} + 4 x^{2} = 7 x^{2}

, but

3 x^{2} + 4 x

stays split.

What is

de g (2 x^{3} + 3 x^{2} - 5 x + 1)

and what does that number measure?

3

— the highest power present, our single measure of "size".

Rewrite

x + 5

in the form

x - c

and give

c

x - (- 5)

, so

c = - 5

(the value that makes the divisor zero).

Compute

\frac{6 x ^{4}}{x}

using the subtract-exponents rule.

6 x^{4 - 1} = 6 x^{3}

State the master division equation and the stopping condition.

P (x) = D (x) Q (x) + R (x)

with

de g (R) < de g (D)

R = 0