[Common Math 1 Part 4] Polynomial Multiplication and the Distributive, Commutative, Associative Laws

Today’s goal:

Understand polynomial multiplication precisely and see, in one sweep, why the commutative, associative, and distributive laws all hold.

• When multiplying polynomials we multiply every pair of terms and then collect terms with the same degree.
• Looking through the lens of arrays (convolution) lets us visualize that workflow.

1. Definition of Polynomial Multiplication

Two steps, always:

1. Multiply each term of one polynomial with all terms of the other.
2. Combine terms that share the same degree.

Example:

$$(2x^2 + 3x + 1)(x + 4)$$

	$x$	$4$
$2x^2$	$2x^3$	$8x^2$
$3x$	$3x^2$	$12x$
$1$	$x$	$4$

Gather equal degrees to get $2x^3 + 11x^2 + 13x + 4$.

2. Why It Feels Like Long Multiplication

Think about multiplying $231 \times 14$:

• Compute $231 \times 4$.
• Compute $231 \times 10$ (shift one place).
• Add the aligned digits.

Polynomials behave the same way:

• $(2x^2 + 3x + 1) \times 4$
• $(2x^2 + 3x + 1) \times x$ (shifts the degree by 1)
• Add terms of the same degree.

So

• Integer multiplication adds matching place values.
• Polynomial multiplication adds matching degrees.

3. Array View (Convolution)

Write polynomials as coefficient arrays:

• $2x^2 + 3x + 1 \rightarrow [2, 3, 1]$
• $x + 4 \rightarrow [1, 4]$

Multiplication now looks like:

• $[2, 3, 1] \times 4 = [8, 12, 4]$
• $[2, 3, 1] \times x = [2, 3, 1, 0]$ (shift one slot)
• Add index by index → $[2, 11, 13, 4]$

Imagine $P = [2, 3, 1]$ as a stamp:

• Each coefficient of $Q$ tells you how strongly to press the stamp.
• The degree in $Q$ tells you how far to shift the stamp to the right.

Every time we look at a term from $Q$ we:

1. Multiply every coefficient of $P$ by that term’s coefficient.
2. Shift the block according to the term’s degree.
3. Overlay it with previously stamped blocks and add the overlapping positions.

This “shift-and-add” perspective is exactly convolution. Nothing about polynomials changes—we simply translate “collect equal degrees” into “add the same array slots.”

See the stages below.

4. Looking at the $i$-th Slot Reveals the Laws

Let

• $P = [p_0, p_1, p_2, \dots]$
• $Q = [q_0, q_1, q_2, \dots]$

and $R = P * Q = [r_0, r_1, r_2, \dots]$. Then

• $r_0 = p_0 q_0$
• $r_1 = p_0 q_1 + p_1 q_0$
• $r_2 = p_0 q_2 + p_1 q_1 + p_2 q_0$
• $r_3 = p_0 q_3 + p_1 q_2 + p_2 q_1 + p_3 q_0$

In general, $r_i$ adds up all pairs whose indices sum to $i$.

Example with $P(x) = 2x^2 + 3x + 1, \quad Q(x) = x + 4$ so $P = [2, 3, 1]$, $Q = [1, 4]$, and $R = [2, 11, 13, 4]$.

Focus on $r_1$:

• $(0, 1)$ → $p_0 q_1 = 2 \cdot 4 = 8$
• $(1, 0)$ → $p_1 q_0 = 3 \cdot 1 = 3$

Only these two pairs sum to 1, so $r_1 = 11$. Translating back, both products contribute to the $x^2$ term: $8x^2 + 3x^2 = 11x^2$.

Now we can prove all three laws by zooming in on the $i$‑th entry.

4-1, 4-2, 4-3 details (commutative/associative/distributive)

4‑1. Commutative Law

$P * Q = Q * P$

The $i$‑th element of $P * Q$ is $r_i = p_0 q_i + p_1 q_{i-1} + \cdots + p_i q_0,$ while the $i$‑th element of $Q * P$ is $r'_i = q_0 p_i + q_1 p_{i-1} + \cdots + q_i p_0.$

Each term is the same product with swapped order, so $r_i = r'_i$ for all $i$.

4‑2. Associative Law

$(P * Q) * T = P * (Q * T)$

Looking at the $i$‑th slot, both sides add exactly the products $p_a q_b t_c$ whose indices add up to $i$. The grouping only changes which partial sums you compute first; the set of terms landing in slot $i$ is identical, so the results match.

4‑3. Distributive Law

$P * (Q + T) = P * Q + P * T$

At index $i$ we have

$$\bigl(P * (Q + T)\bigr)_i = p_0 (q_i + t_i) + p_1 (q_{i-1} + t_{i-1}) + \cdots.$$

Distribute term by term to get the $i$‑th slot of $P * Q$ plus the $i$‑th slot of $P * T$.

5. Practice Quiz: Single-Variable Multiplication

Use the quiz below to

• Read the prompt
• Enter coefficients in descending order
• Walk through the partial products

6. Extending to Two Variables

The same idea applies:

• 1 variable → 1‑D coefficient array
• 2 variables → 2‑D coefficient array (matrix)
• Rules → multiply coefficients, add exponents (row/column indices)

6‑1. Example With Matrices

Let $P(x, y) = x + y$, $Q(x, y) = x - y$. Start with a zero matrix for $R$ and add contributions via

$$R[r + u,\; c + v] \mathrel{+}= Q[r, c] \cdot P[u, v].$$

In words: for every nonzero coefficient in $Q$, create a stamp of $P$, scale it, shift it, and overlay it. That is 2‑D convolution.

The final polynomial is $(x + y)(x - y) = x^2 - y^2$ because the $xy$ terms cancel.

Try it interactively:

7. Key Takeaways

• Polynomial multiplication = “multiply terms, then collect equal degrees.”
• It mirrors long multiplication, so the process is intuitive.
• The array/convolution view is simply a clearer window into the same rule.
• Inspecting the $i$‑th slot makes the commutative, associative, and distributive laws self-evident.

In short:

Polynomial multiplication is not a new rule—it is the systematic execution of “distribute, then collect like degrees.”