[Common Math 1 Part 6] Polynomial Division and Its Radix Connection

In this post, we learn polynomial division in two connected ways:

First with long division, then with a coefficient-array view that keeps the same logic but hides the variables, and finally by stepping back to connect all polynomial operations to radix structure.

We work with polynomials in one variable and real-number coefficients, and we assume the divisor is not the zero polynomial.

Just like integer division, polynomial division repeatedly subtracts a suitable multiple of the divisor. The stopping rule is different:

• In integer division, we stop when the remainder is smaller than the divisor in value.
• In polynomial division, we stop when the remainder has lower degree than the divisor.

Here, the degree means the highest exponent of the variable, and a coefficient is the number attached to each power of $x$.

Keep these three ideas in mind:

• We cancel the highest-degree term first.
• Missing powers must be written with coefficient 0.
• The array view is the same calculation written only with coefficients.

1. Fundamentals of Polynomial Division

1‑1. Why Bother With Division?

Same reasons as with numbers:

• Factorization: ** $(x^2 - 5x + 6) \div (x - 2) = x - 3$ ⇒ $x^2 - 5x + 6 = (x - 2)(x - 3)$
• Solving equations: ** if $x = 1$ is a root of $x^3 - 6x^2 + 11x - 6 = 0$, divide by $(x - 1)$ to lower the degree.
• Analyzing rational functions: ** study $\frac{P(x)}{Q(x)}$ by splitting it into quotient + remainder.

1‑2. Definition

If we divide a polynomial $A(x)$ by a nonzero polynomial $B(x)$, then there are unique polynomials $Q(x)$ and $R(x)$ such that

$$A(x) = B(x) \cdot Q(x) + R(x)$$

where $R(x)=0$ or $\deg R < \deg B$.

In plain English:

• $A(x)$ is the dividend.
• $B(x)$ is the divisor.
• $Q(x)$ is the quotient.
• $R(x)$ is the remainder.

So polynomial division has the same overall form as integer division. The only difference is that the remainder is controlled by degree, not by size.

Example: $(2x^3 - 3x + 1) \div (x - 2)$

• Dividend: $2x^3 - 3x + 1$
• Divisor: $x - 2$
• Quotient: $2x^2 + 4x + 5$
• Remainder: $11$

1‑3. Column-style Division

Think about integer long division first, say $231 \div 11$.

1. $23 \div 11 = 2$ → place 2 in the quotient.
2. Subtract $2 \times 11 = 22$.
3. Bring down the next digit and repeat.

Polynomial long division follows the same pattern:

1. Divide the highest-degree term of the current dividend by the highest-degree term of the divisor.
2. Write that term in the quotient.
3. Multiply the divisor by that term and subtract.
4. Repeat with the new remainder until the remainder’s degree is smaller than the divisor’s degree.

For $(2x^3 - 3x + 1) \div (x - 2)$ we rewrite the dividend as $2x^3 + 0x^2 - 3x + 1$.

That inserted $0x^2$ is not a new term. It is only a placeholder so each degree has its own column.

Now the long division works like this:

1. Divide the leading terms: $2x^3 \div x = 2x^2$. Put $2x^2$ in the quotient.
2. Multiply the divisor by $2x^2$: $(x - 2)(2x^2)=2x^3-4x^2$.
3. Subtract. The new expression is $4x^2 - 3x + 1$.
4. Repeat: $4x^2 \div x = 4x$, so the next quotient term is $4x$.
5. Multiply and subtract again: $(x - 2)(4x)=4x^2-8x$, leaving $5x+1$.
6. Repeat once more: $5x \div x = 5$, so the last quotient term is $5$.
7. Multiply and subtract: $(x - 2)(5)=5x-10$, leaving remainder $11$.

The usual column layout is

$$\begin{array}{r|rrrr} \text{Quotient} & & 2x^2 & +4x & +5 \\ \hline x-2 & 2x^3 & +0x^2 & -3x & +1 \\ & 2x^3 & -4x^2 & & \\ & & 4x^2 & -3x & \\ & & 4x^2 & -8x & \\ & & & 5x & +1 \\ & & & 5x & -10 \\ & & & & 11 \\ \end{array}$$

Essentials:

• Align like degrees vertically.
• Fill missing degrees with zero coefficients so nothing slips.
• After each subtraction, the leading degree gets smaller.

Finally, check the result:

$$2x^3 - 3x + 1 = (x - 2)(2x^2 + 4x + 5) + 11$$

This check matters because polynomial division is not finished until the identity works.

2. Division Through Arrays

2‑1. Why Arrays Help

The long-division steps do not really depend on the letter $x$. They depend on the order of the coefficients.

So if the powers are written in descending order and missing powers are filled with 0, we can record the same polynomial as a list of coefficients.

Polynomial form	Array form	Benefit
$2x^3 + 0x^2 - 3x + 1$	[2, 0, -3, 1]	Missing degrees stay visible
$x - 2$	[1, -2]	Focus on coefficients
Degree tracking	Indices = degrees	Ideal for computation

This does not create a different theorem. It is the same division process, just written more compactly.

2‑2. Array Algorithm

Think of the division as progressively removing the leading coefficient from left to right:

Step	Action
0	Start with dividend [2, 0, -3, 1] and divisor [1, -2].
1	Leading coefficients: $2 \div 1 = 2$. So the first quotient coefficient is 2. Multiply [1, -2] by 2 to get [2, -4], then subtract it from the first two entries. The new row becomes [0, 4, -3, 1].
2	Ignore the leading 0. Now $4 \div 1 = 4$, so the next quotient coefficient is 4. Multiply [1, -2] by 4 to get [4, -8], then subtract from the next two entries. The new row becomes [0, 0, 5, 1].
3	Ignore the next leading 0 again. Now $5 \div 1 = 5$, so the last quotient coefficient is 5. Multiply [1, -2] by 5 to get [5, -10], then subtract. The final row becomes [0, 0, 0, 11].
Result	Quotient [2, 4, 5], which means $2x^2 + 4x + 5$, and remainder $11$.

This mirrors the column method exactly. The only difference is that the powers of $x$ are being tracked by position.

For divisors of the form $(x-c)$, this coefficient view later becomes the basis for synthetic division.

2‑3. Worked Examples

Example 1: Clean division

$$(x^2 + 3x + 2) \div (x + 1)$$

Array: [1, 3, 2] ÷ [1, 1]

• Quotient [1, 2] → $x + 2$
• Remainder $0$

Example 2: Nonzero remainder

$$(x^2 + 2x + 5) \div (x + 1)$$

Array: [1, 2, 5] ÷ [1, 1]

• Quotient [1, 1] → $x + 1$
• Remainder $4$

Example 3: Another quadratic

$$(2x^2 + 5x + 3) \div (x + 1) = 2x + 3,$$

remainder $0$.

Example 4: Cubic

$$(x^3 - 6x^2 + 11x - 6) \div (x - 1) = x^2 - 5x + 6,$$

remainder $0$, so the cubic factors as $(x - 1)(x - 2)(x - 3)$.

Notice that when the divisor is $(x-a)$, the remainder also equals the value of the polynomial at $x=a$. For our main example,

$$P(2)=2(2^3)-3(2)+1=16-6+1=11$$

which matches the remainder. We will study this idea more directly in the next post.

Experiment with the visualizer to see each subtraction step unfold:

3. Looking Back: Structural Resemblance Between Polynomial Operations and Radix Arithmetic

Now that we have seen addition, subtraction, multiplication, and division across the series, we can step back and explain why polynomial operations often feel similar to calculations in base 2, base 10, or base 16. The goal here is not to claim that polynomials and radix numerals are the same object, but to use a familiar place-value picture to reflect on the structure we have learned.

Let us draw the boundary clearly first.

Warning: this comparison is a structural analogy. Polynomials and radix numerals are different mathematical objects, and carry/borrow rules from radix arithmetic do not transfer directly to polynomial algebra.

Even the division algorithm we just studied shows the resemblance.

• Integer long division chooses the next quotient digit from the highest place, multiplies, subtracts, and brings down the next part.
• Polynomial long division chooses the next quotient term from the highest degree, multiplies, subtracts, and continues with the lower-degree remainder.

Across all four operations, the common thread is that we work through an ordered sequence of places.

The core idea is simple:

$$3x^2 + 2x + 5$$

means “multiply each place by a power of $x$ and add.” That is the same skeleton used in place-value notation.

Expression	Place-value expansion
$325_{10}$	$3\cdot 10^2 + 2\cdot 10 + 5$
$1011_2$	$1\cdot 2^3 + 0\cdot 2^2 + 1\cdot 2 + 1$
$2A_{16}$	$2\cdot 16 + 10$
$3x^2 + 2x + 5$	$3\cdot x^2 + 2\cdot x + 5$

This is why a polynomial can remind us of a formal base-x expression. But this is only a structural analogy, not an equivalence.

There is still one important difference:

• In a polynomial, coefficients are free, and there is no carrying rule.
• In a base-$b$ numeral, each digit must stay between 0 and b-1.

So carrying and borrowing are not basic polynomial rules. They are procedures required in radix notation after we fix a base and rewrite the result with valid digits.

3.1 Addition and Subtraction: Match Equal Places

Polynomial addition combines coefficients of the same degree. Radix addition combines digits of the same place value. The skeleton is the same.

For example,

$$101_2 + 011_2$$

becomes

$$(x^2 + 1) + (x + 1) = x^2 + x + 2.$$

Up to this point, it is ordinary polynomial addition. Then we set x=2 and apply binary digit rules, so we rewrite the result as the binary numeral $1000_2$.

Borrowing is the reverse idea:

• In base 10, one ten becomes ten ones.
• In base 2, one two becomes two ones.
• In base 16, one sixteen becomes sixteen ones.

So both addition and subtraction follow the pattern “match equal places first.” The extra carrying or borrowing step belongs only to radix notation.

3.2 Multiplication: Collect Like Degrees vs. Collect Like Places

As Part 4 showed, polynomial multiplication multiplies every term and then collects equal degrees. Long multiplication does the same with place values.

For example,

$$11_2 \times 11_2$$

can be read as

$$(x+1)(x+1) = x^2 + 2x + 1.$$

Then with x=2, the middle coefficient 2 must be rewritten according to binary digit rules, so the result becomes $1001_2$.

So:

• Polynomial multiplication: collect like degrees.
• Radix multiplication: collect like places, then carry if needed.

The correspondence becomes even clearer if we place an actual binary long-multiplication example next to the polynomial version.

$$101_2 \times 11_2$$

In binary long multiplication, we compute:

   101_2
x   11_2
--------
   101_2
 1010_2
--------
 1111_2

Now read the same digits as a polynomial:

$$(x^2 + 1)(x + 1)$$

and expand:

   x^2 + 1
x    x + 1
-----------
   x^2 + 1
 x^3 + x
-----------
 x^3 + x^2 + x + 1

So we get the direct correspondences:

• 101_2 ↔ $x^2 + 1$
• 11_2 ↔ $x + 1$
• shifting each partial product left ↔ increasing degree by 1

The parallel becomes even clearer in a compact comparison table:

Binary multiplication	Polynomial multiplication
101_2 × 1 = 101_2	$(x^2 + 1) \times 1 = x^2 + 1$
101_2 × 10_2 = 1010_2	$(x^2 + 1) \times x = x^3 + x$
101_2 + 1010_2 = 1111_2	$(x^2 + 1) + (x^3 + x) = x^3 + x^2 + x + 1$

Finally, when we set x=2,

$$x^3 + x^2 + x + 1 \rightarrow 1111_2,$$

which matches the actual binary multiplication result exactly.

The key point here is the parallel procedure.

• In binary, shifting one place left means multiplying by 2 once more.
• In polynomials, increasing the degree by one means multiplying by x once more.

So both calculations follow the same flow: multiply, shift, then combine equal places/degrees. The difference is that binary arithmetic may need carrying at the end, while polynomials keep each degree as it is.

3.3 Division: Long Division for Degrees and for Digits

This post adds the last piece: division. In integer long division, we choose the next quotient digit from the highest place, multiply, subtract, and continue. Polynomial long division does the same with highest-degree terms.

The comparison becomes especially vivid with a concrete binary division example. Consider

$$1101_2 \div 11_2.$$

In binary division, we reason like this:

• 11_2 fits once into the leading 11_2, so the first quotient digit is 1.
• Subtract 11_2, giving 0, and bring down the next digit.
• At the next place, 11_2 does not fit, so the next quotient digit is 0.
• At the end, 11_2 still does not fit into 1_2, so the remainder is 1_2.

Therefore,

$$1101_2 \div 11_2 = 100_2 \text{ ... } 1_2.$$

Now read the same expression as polynomial division:

$$(x^3 + x^2 + 1) \div (x + 1).$$

With polynomial long division,

• we choose $x^2$ as the first quotient term to eliminate the highest-degree term $x^3$;
• subtract $x^2(x+1)=x^3+x^2$;
• the remainder becomes $1$;
• since the remainder has lower degree than $(x+1)$, we stop.

So

$$(x^3 + x^2 + 1) \div (x + 1) = x^2 \text{ ... } 1.$$

We can even verify the result in parallel:

• Binary: $11_2 \times 100_2 + 1_2 = 1100_2 + 1_2 = 1101_2$
• Polynomial: $(x+1)\cdot x^2 + 1 = x^3 + x^2 + 1$

Placed side by side, the matching parts are:

Binary division	Polynomial division
$1101_2$	$x^3 + x^2 + 1$
$11_2$	$x + 1$
quotient $100_2$	quotient $x^2$
remainder $1_2$	remainder $1$

The step-by-step comparison is:

Step	Binary	Polynomial
1	11_2 fits once into the leading 11_2	choose $x^2$ to cancel the highest-degree term $x^3$
2	multiply, subtract, and bring down the next digit	multiply, subtract, and continue with the lower-degree remainder
3	stop when 11_2 no longer fits into 1_2	stop when the remainder degree is lower than the divisor degree

So both algorithms follow the same backbone: choose the next quotient piece from the highest place/highest degree, multiply, subtract, and stop when division can no longer continue.

The real lesson of this comparison is not that numbers and polynomials are identical. It is that the long-division algorithm itself works in both a place-value system and a degree-based system.

Both algorithms stop when the remainder is "small enough," but the meaning of that phrase is different:

• Integer division stops when the remainder is smaller than the divisor in value.
• Polynomial division stops when the remainder has lower degree than the divisor.

So numbers are controlled by size, while polynomials are controlled by degree. The skeleton of the algorithm is similar, but the stopping criteria measure different properties.

3.4 Put Together in One Table

Operation	Polynomial view	Radix view	Key difference
Addition	Add coefficients of equal degree	Add equal place values	Radix notation may require carrying
Subtraction	Subtract coefficients of equal degree	Subtract equal place values	Radix notation may require borrowing
Multiplication	Multiply terms, then collect like degrees	Multiply digit blocks, then collect like places	Radix notation then rewrites the result with valid digits
Division	Remove highest degrees first by long division	Remove highest places first by long division	The stopping criterion uses degree vs. magnitude

From this perspective, a polynomial is not a completely alien object. It has a structure that resembles the place-value systems students already know, even though the rules are not identical.

4. Summary

4‑1. Essence of Polynomial Division

• Systematically eliminate the highest-degree term.
• Array view = operate on matching indices.
• Always verify $A = B \cdot Q + R$ and check that the remainder has lower degree than the divisor.

4‑2. Meaning of the Array View

Polynomial        → Row of coefficients ordered by degree
Long division     → Remove higher degrees first
Array notation    → Record those steps numerically
Quotient/remainder→ Structured output of the same process

4‑3. Toward the Next Topic

Here we learned the general division algorithm. Next we will focus on divisors of the form $(x-c)$ and see why the remainder can be found from the value $P(c)$. That idea leads naturally to the remainder theorem, factor theorem, and then to synthetic division.

In short:

Polynomial division is a systematic degree-reduction process, and the array viewpoint exposes how the coefficients flow.