[Common Math 1 Part 3] Bivariate Polynomials and Matrices

1. From One Variable to Two

1.1 Why Do We Need Two Variables?

Many real-world phenomena depend on multiple variables. Whenever a single variable fails to capture the relationship, a bivariate polynomial becomes our tool.

Bivariate polynomials formalize these multidimensional relationships.

1.2 Anatomy of a Bivariate Polynomial

Consider $P(x, y) = 3x^2y + 2xy^2 + 5x + 7y + 1.$

This table gives us the information we need to place each term in a grid.

Degree terminology:

• The degree in $x$ is the largest exponent of $x$ that appears, so here it is 2.
• The degree in $y$ is the largest exponent of $y$ that appears, so here it is 2.
• The total degree of one term is the sum of its exponents. For example, $x^2y$ has total degree $2 + 1 = 3$.
• The total degree of the polynomial is the largest total degree among its terms, so here it is 3.

We will use the degree of $y$ for rows and the degree of $x$ for columns. That choice is a convention, not a law, but once we choose it we must use it consistently.

2. Array Representation

2.1 Thinking in 2‑D Arrays

Use the degree of $y$ for rows and the degree of $x$ for columns. To make that visible, start with a labeled coefficient table.

Each position stores the coefficient of exactly one term:

• row $y^2$, column $x^1$ gives 2, so that entry represents $2xy^2$
• row $y^1$, column $x^2$ gives 3, so that entry represents $3x^2y$
• row $y^0$, column $x^1$ gives 5, so that entry represents $5x$
• row $y^0$, column $x^0$ gives 1, so that entry represents the constant term

The same information can be written as a matrix:

Because the degrees are listed in descending order, the top row is the $y^2$ row and the leftmost column is the $x^2$ column.

Reading the matrix:

• top middle entry 2 means the coefficient of $x^1y^2$
• middle left entry 3 means the coefficient of $x^2y^1$
• bottom middle entry 5 means the coefficient of $x^1y^0$
• bottom right entry 1 means the coefficient of $x^0y^0$

This row-column layout is a convention. We could swap the roles of $x$ and $y$, but then every example and operation would need to follow that new convention consistently.

Visualization:

2.2 Seeing Polynomials as Matrices

Why matrices?

1. Structural match: rows/columns correspond directly to the two degrees.
2. Addition match: matrix addition exactly matches polynomial addition coefficient by coefficient.
3. Clear bookkeeping: even when some terms are missing, their positions still stay fixed in the grid.

P = np.array([
    [0, 2, 0],
    [3, 0, 7],
    [0, 5, 1]
])

print(P.shape)
print(P[1, 0])  # coefficient of x²y
print(P[0, 1])  # coefficient of xy²

3. Operating on Bivariate Polynomials

3.1 Why Addition Works

Suppose two polynomials both contain an $x^ay^b$ term. When we add the polynomials, we add the coefficients of those matching terms.

That is exactly what the matrix does. The entry in one fixed position always represents one fixed degree pair, so adding two matrices entry by entry is the same as adding two polynomials term by term.

For example, in the matrix below the middle entry of the bottom row represents the coefficient of $x$:

If another polynomial has 4 in that same position, then the new coefficient of $x$ becomes $5 + 4 = 9$.

3.2 Matrix Addition

Let

$P(x, y) = 2x^2 + xy + 3y + 1$

$Q(x, y) = x^2 - xy + 2y^2 + 4$

Their matrices are

Element-wise addition gives

which means $3x^2 + 2y^2 + 3y + 5$.

Notice what happened to the $xy$ term: the coefficient 1 in $P$ and the coefficient -1 in $Q$ are in the same position, so they add to 0 and disappear.

P = np.array([
    [0, 0, 0],
    [0, 1, 3],
    [2, 0, 1]
])

Q = np.array([
    [0, 0, 2],
    [0, -1, 0],
    [1, 0, 4]
])

print(P + Q)

3.3 Real Applications

Image processing

• RGB images store brightness as 2‑D arrays.
• Filters use nearby entries in the array to blur or sharpen an image.

# Simplified brightness adjustment
image = np.array([...])
brightness_adjusted = image + 20

Physics simulations

• Model heat or pressure at each grid point $(x, y)$.

Economics

• Model cost as a function of two resources and study how the cost changes.

3.4 Practice Problems

Extra drills

Problem 1 Express $P(x, y) = 2x^2 + 3xy - y^2 + 5x - 2y + 4$ as a matrix.

Problem 2 Let $Q(x, y) = x^2 - xy + 2y^2 - 3x + y - 1$. Find $P + Q$ via matrix addition.

🎯 Hands-on practice

4. Wrap-Up

Extending to two variables multiplies our expressive power. We can move from straight-line worlds to curved surfaces and coupled relationships.

Key takeaways:

• Bivariate polynomials model multidimensional relationships.
• Matrices give a systematic way to manage the coefficients.
• Matrix addition matches polynomial addition because each position represents one degree pair.

This foundation leads directly to linear algebra, computer graphics, machine learning, and more.

Next preview: extending to three, four, or even $n$ variables introduces higher-dimensional arrays—tensors—which power modern AI and deep learning. Stay tuned!