Polynomials

A term is a number, a variable, or a number multiplied by variables raised to whole-number powers. Examples of terms: $7$, $3x$, $-5x^2$, $x^3$.

A polynomial is a sum of terms. In each term:

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• the coefficient is the number multiplied by the variable (in $-5x^2$ the coefficient is $-5$);
• a constant is a term with no variable (like $2$ in $3x^2-5x+2$).

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• Monomial — one term: $4x^3$
• Binomial — two terms: $x+3$
• Trinomial — three terms: $3x^2-5x+2$
• Polynomial — the general word for one or more terms (four or more terms usually just get called “a polynomial”).

The degree of a term is the exponent on its variable ($-5x^3$ has degree $3$; a constant like $7=7x^0$ has degree $0$).

The degree of a polynomial is the largest degree among its terms.

A polynomial is in standard form when its terms are written in descending order of degree (highest power first).

The leading coefficient is the coefficient of the first term once the polynomial is in standard form (the coefficient of the highest-degree term).

To evaluate a polynomial for a given value, replace every variable with that value (use parentheses!), then follow the order of operations.

Like terms have the exact same variable part (same variable, same exponent), such as $3x^2$ and $5x^2$. To add polynomials, drop the parentheses and combine like terms.

To subtract, distribute the minus sign to every term of the second polynomial (this flips each sign), and then combine like terms.

Multiply the monomial by each term inside the parentheses. Multiply the coefficients, and add the exponents on matching variables ($x^a\cdot x^b=x^{a+b}$).

FOIL names the four products you make when multiplying two binomials:

Then combine the two middle (Outer $+$ Inner) like terms.

Multiply each term of the binomial by each term of the trinomial (that is $2\times 3 = 6$ small products), then combine like terms and write in standard form.

These come straight from FOIL, but recognizing the pattern saves time.

FOIL and repeated distribution get slow for high powers. The Binomial Theorem expands $(a+b)^n$ all at once:

where the binomial coefficient $\binom{n}{k} = \dfrac{n!}{k!\,(n-k)!}$ counts the ways to choose $k$ of the $n$ factors that contribute a $b$.

Reading the pattern for a fixed $n$: the power of $a$ starts at $n$ and drops by one each term, the power of $b$ starts at $0$ and rises by one, and the two exponents in every term add up to $n$.

The coefficients $\binom{n}{k}$ are exactly the rows of Pascal's triangle. Each entry is the sum of the two entries above it, and every row begins and ends with $1$.

So $(a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$: the row $1,4,6,4,1$ becomes the coefficients directly.

Sometimes you only need a single coefficient of a product. Instead of multiplying everything out, collect only the pairs of terms whose degrees add up to the power you want.

To find the coefficient of $x^k$ in a product, list every way to pick one term from each factor so their exponents sum to $k$, multiply those coefficients, and add the results.

The sum $a+b$ and product $ab$ contain enough information to rebuild higher symmetric expressions. Starting from $(a+b)^2 = a^2+2ab+b^2$ and $(a+b)^3 = a^3+3a^2b+3ab^2+b^3$, rearrange:

The second identity also factors as $a^3+b^3=(a+b)(a^2-ab+b^2)$; both give the same value.

The perfect-square pattern generalizes to three terms. Every term squares, and every distinct pair produces a doubled cross term:

There are three squares and three cross terms because there are three ways to pair up $a$, $b$, and $c$.

For large $|x|$, a polynomial behaves like its leading term alone---the highest-degree term dominates every other. So the degree and the sign of the leading coefficient decide the end behavior:

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• Even degree: both ends point the same way---up/up if the leading coefficient is positive, down/down if negative.
• Odd degree: the ends point opposite ways---down/up if positive, up/down if negative.

When a polynomial does not divide evenly, long division splits it into a quotient plus a remainder, mirroring long division of numbers. At each step: divide the leading term of the current dividend by the leading term of the divisor, multiply back, and subtract.