7. Polynomial Operations

Polynomial Operations

A polynomial is a sum of terms of the form $a \cdot x^n$ where $n$ is a non-negative integer. The degree is the highest power; the leading coefficient is the coefficient of that term.

Adding and subtracting

Combine like terms: terms with the same variable raised to the same power.

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

When subtracting, distribute the negative to every term in the second polynomial:

$(4x^2 + 3x - 2) - (x^2 - x + 5) = 3x^2 + 4x - 7$

The $-$ flips signs on $x^2$, $-x$, and $5$.

Multiplying

Use the distributive property. For two binomials, FOIL is a useful mnemonic:

• First: First terms of each binomial
• Outer: Outer pair
• Inner: Inner pair
• Last: Last terms of each binomial

$(x + 3)(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15$

For larger polynomials, use distribution systematically: every term in the first polynomial multiplied against every term in the second.

Special products (memorize these)

The middle term in $(a \pm b)^2$ is the most common point of error: students forget the $2ab$ and write $a^2 + b^2$ instead.

Connection to ML

ML loss functions are polynomials in the parameters. Mean Squared Error is a degree-2 polynomial in each $\theta_j$. Polynomial regression uses degree-$d$ features ($x$, $x^2$, $xy$, $y^2$, …). The characteristic polynomial of an $n \times n$ matrix is degree $n$, and its roots are the matrix's eigenvalues. Polynomial fluency is the algebra layer beneath nearly every ML linear-algebra computation you'll do.