The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 10 (Intermediate Algebra) Adding, Multiplying, and Dividing Polynomials

Overview

This lecture covers the three core operations on polynomials: addition (combining like terms), multiplication (distribution), and division (breaking a fraction into pieces, plus polynomial long division). It opens by naming the building blocks — terms, and polynomials classified by how many terms they have. Adding polynomials reduces to collecting like terms; multiplying relies entirely on careful distribution followed by combining like terms; and division splits into two cases: dividing by a single term (split the fraction termwise and cancel) versus dividing by a multi-term polynomial (long division with a possible remainder). Two special-product patterns — the difference of squares and the square of a binomial — are introduced as shortcuts you can always fall back to brute-force distribution to verify.

Key Concepts

Term — a coefficient together with some variables: $14x^2y$, $2x$, $12y^2$, $7$, and $y$ are all terms.

Polynomial — several terms added together. Named by term count:

• Monomial — one term (e.g. $7x$).
• Binomial — two terms (e.g. $4 + 2x$, or $4x^2y - 17z$).
• Trinomial — three terms.

Like terms — terms with the same variables, each variable to the same power. Only like terms can be combined, and you combine them by adding their coefficients. Because multiplication is commutative, $xy^3 = y^3x$, so those count as the same term. By convention, write variables in alphabetical order: $3y^3x$ is written $3xy^3$.

Adding/subtracting polynomials — collect like terms. When subtracting a polynomial, first distribute the negative sign through every term inside its parentheses, then combine.

Multiplying polynomials — distribute every term of the first factor to every term of the second, then combine like terms. When multiplying individual terms, multiply coefficients and add exponents of matching variables (recall $x \cdot x = x^2$). For three or more factors, the order does not matter (multiplication is associative): multiply two factors first, then multiply that result by the next — do not "over-distribute" a factor more than once.

Special product — difference of squares:

The two middle terms ($+ab$ and $-ab$) cancel.

Special product — square of a binomial:

These are shortcuts; you can always multiply out by hand to confirm.

Dividing by a single term (monomial) — split the fraction into one fraction per numerator term over the common denominator, then simplify each (divide coefficients, subtract exponents). Note that a term divided by itself equals $1$, not $0$.

Polynomial long division (dividing by a multi-term polynomial) — works exactly like integer long division:

• Write both polynomials in decreasing order of exponent, inserting a $+0x^k$ placeholder for any missing power so the columns line up.
• Divide the leading term of the dividend by the leading term of the divisor, multiply the result back through the divisor, subtract, bring down the next term, and repeat.
• The most common mistake is forgetting to flip the signs when subtracting.
• Stop when the remaining degree is below the divisor's degree; write any leftover as $+\dfrac{\text{remainder}}{\text{divisor}}$. A remainder of $0$ means the divisor divides evenly.

Worked Example

Multiplying two binomials — distribute, then combine like terms:

Polynomial long division — divide $3x^3 - 5x^2 - 32x + 7$ by $x - 4$, using the symbolic step "divide leading term, multiply back, subtract":

Summary

• A term is a coefficient with variables; a polynomial is terms added together, named monomial / binomial / trinomial by term count.
• Add polynomials by combining like terms (same variables, same powers); when subtracting, distribute the negative through the whole polynomial first.
• Multiply by distributing every term to every term and combining like terms; multiplying terms means multiplying coefficients and adding exponents.
• Memorize the two special products as shortcuts: $(a+b)(a-b) = a^2 - b^2$ and $(a+b)^2 = a^2 + 2ab + b^2$ — but you can always brute-force the distribution.
• Divide by a monomial by splitting the fraction termwise and simplifying each piece (remember $\tfrac{a}{a} = 1$).
• Divide by a multi-term polynomial with long division: order terms by descending degree, insert $0$ placeholders for missing powers, and express any leftover as a remainder over the divisor.