1. What a polynomial is
An algebraic expression of the form $a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where the $a_i$ are constant coefficients and the exponents are non-negative whole numbers. Each $a_i x^i$ is a term; the largest exponent $n$ with $a_n \neq 0$ is the polynomial's degree.
Examples:
- $3x + 5$ — degree $1$. A linear polynomial.
- $x^2 - 4x + 7$ — degree $2$. A quadratic.
- $2x^3 + x - 1$ — degree $3$. A cubic.
- $7$ — a polynomial of degree $0$, just a constant.
And non-examples:
- $\tfrac{1}{x}$ — has a variable in the denominator (i.e., a negative exponent). Not a polynomial.
- $\sqrt{x}$ — has a fractional exponent. Not a polynomial.
- $2^x$ — variable in the exponent. Not a polynomial; this is an exponential function.
Special names by number of terms:
- Monomial — one term: $3x^2$, $7$, $-x$.
- Binomial — two terms: $x + 1$, $2x^2 - 5$.
- Trinomial — three terms: $x^2 + 3x + 2$.
- Past three, "polynomial" is the catch-all term.
The leading coefficient is the coefficient of the highest-degree term. In $4x^3 - 2x + 7$, the leading coefficient is $4$. Standard form writes the terms in descending order of degree, so the leading term appears first.
2. Adding and subtracting
Same rule as for any expression: combine like terms. Terms are alike when their variable parts (including exponents) match.
$$ (3x^2 + 2x + 5) + (x^2 - 4x + 1) = 4x^2 - 2x + 6. $$For subtraction, distribute the minus sign across every term in the second polynomial, then combine like terms:
$$ (3x^2 + 2x + 5) - (x^2 - 4x + 1) = 3x^2 + 2x + 5 - x^2 + 4x - 1 = 2x^2 + 6x + 4. $$Notice that $-(-4x) = +4x$ and $-(+1) = -1$. The minus sign hits every term inside the parentheses, not just the first.
3. Multiplying polynomials
Multiplication is distributivity, applied repeatedly. Every term of the first polynomial multiplies every term of the second; then you combine like terms.
Monomial times polynomial
$$ 3x(2x^2 - 4x + 5) = 6x^3 - 12x^2 + 15x. $$Just distribute the $3x$ across each term.
Binomial times binomial: FOIL
The mnemonic FOIL — First, Outer, Inner, Last — names the four pairings when multiplying two binomials:
$$ (x + 3)(x + 5) = \underbrace{x \cdot x}_{F} + \underbrace{x \cdot 5}_{O} + \underbrace{3 \cdot x}_{I} + \underbrace{3 \cdot 5}_{L} = x^2 + 5x + 3x + 15 = x^2 + 8x + 15. $$FOIL is just full distribution, named so beginners don't forget any of the four products. For larger polynomials, the principle is the same — every term meets every term — but the mnemonic stops being useful.
General products
$$ (x + 2)(x^2 - x + 3) = x \cdot x^2 - x \cdot x + 3x + 2 \cdot x^2 - 2x + 6 = x^3 + x^2 + x + 6. $$Three terms from the first parens, three from the second. Three times two terms multiplies out to six products; combine like terms to finish.
4. Factoring
This page introduces factoring as an idea; the full toolbox — GCF, grouping, difference of squares, perfect-square trinomials, sum/difference of cubes, and the AC method — gets its own treatment in the next topic, Factoring Techniques.
Factoring is multiplication in reverse: given a polynomial, find the product that gives it. It's the harder direction — multiplication is mechanical, factoring is often a search. Three techniques cover most introductory cases.
Greatest common factor (GCF)
If every term shares a common factor, pull it out:
$$ 6x^3 + 9x^2 - 12x = 3x(2x^2 + 3x - 4). $$The GCF is $3x$ (the largest constant common to all coefficients times the lowest power of $x$ present in all terms). Always try this first — it usually simplifies whatever else you need to do.
Factoring by grouping
Useful for four-term polynomials. Group the terms into pairs, factor each pair, and look for a common binomial factor:
$$ x^3 + 2x^2 + 3x + 6 = x^2(x + 2) + 3(x + 2) = (x^2 + 3)(x + 2). $$Factoring trinomials of the form $x^2 + bx + c$
You're looking for two numbers whose product is $c$ and whose sum is $b$. For $x^2 + 8x + 15$: we need two numbers multiplying to $15$ and adding to $8$. They are $3$ and $5$. So:
$$ x^2 + 8x + 15 = (x + 3)(x + 5). $$Check by expanding: FOIL gives back the original.
When the leading coefficient isn't $1$ (say, $2x^2 + 7x + 3$), the procedure is similar but a touch more involved — find two numbers that multiply to $2 \cdot 3 = 6$ and add to $7$ ($1$ and $6$), split the middle term, then factor by grouping: $2x^2 + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1)$.
5. Special patterns
Three patterns appear so often it's worth recognizing them on sight.
Difference of squares
$$ a^2 - b^2 = (a - b)(a + b). $$Verify by FOIL: $(a - b)(a + b) = a^2 + ab - ab - b^2 = a^2 - b^2$. The middle terms cancel.
$x^2 - 25 = (x - 5)(x + 5)$. $4x^2 - 9 = (2x - 3)(2x + 3)$. The pattern: a perfect square minus a perfect square factors as a product of the sum and the difference of the square roots.
Perfect-square trinomials
$$ a^2 + 2ab + b^2 = (a + b)^2, \qquad a^2 - 2ab + b^2 = (a - b)^2. $$$x^2 + 6x + 9 = (x + 3)^2$. The clue is that the middle term is twice the product of the square roots of the first and last terms ($2 \cdot x \cdot 3 = 6x$).
Sum and difference of cubes
$$ a^3 + b^3 = (a + b)(a^2 - ab + b^2). $$ $$ a^3 - b^3 = (a - b)(a^2 + ab + b^2). $$Less commonly used than the previous two but worth knowing for cubic factoring.
When factoring an unfamiliar polynomial: (1) always try GCF first; (2) count terms — two terms suggests difference of squares or cubes; three terms suggests a trinomial; four terms suggests grouping; (3) recognize special patterns by inspection. This sequence catches almost every introductory factoring problem.