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.

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

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

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.